a survey of lagrangian mechanics and control on lie

A SURVEY OF LAGRANGIAN MECHANICS AND CONTROLON LIE ALGEBROIDS AND GROUPOIDS

JORGE CORTES, MANUEL DE LEON, JUAN C. MARRERO, D. MARTIN DE DIEGO,

AND EDUARDO MARTINEZ

Abstract. In this survey, we present a geometric description of Lagrangianand Hamiltonian Mechanics on Lie algebroids. The flexibility of the Lie al-gebroid formalism allows us to analyze systems subject to nonholonomic con-straints, mechanical control systems, Discrete Mechanics and extensions toClassical Field Theory within a single framework. Various examples along thediscussion illustrate the soundness of the approach.

Contents

1. Introduction 22. Lie algebroids and Lie groupoids 42.1. Lie algebroids 42.2. Lie groupoids 83. Mechanics on Lie algebroids 123.1. Lagrangian Mechanics 133.2. Hamiltonian Mechanics 153.3. The Legendre transformation and the equivalence between the

Lagrangian and Hamiltonian formalisms 164. Nonholonomic Lagrangian systems on Lie algebroids 174.1. Constrained Lagrangian systems 174.2. Regularity, projection of the free dynamics and nonholonomic bracket 184.3. Reduction 204.4. Example: a rolling ball on a rotating table 214.5. Hamiltonian formalism 235. Mechanical control systems on Lie algebroids 245.1. General control systems on Lie algebroids 245.2. Mechanical control systems 245.3. Accessibility and controllability notions 265.4. The structure of the control Lie algebra 275.5. Accessibility and controllability tests 286. Discrete Mechanics on Lie groupoids 29

2000 Mathematics Subject Classification. 17B66, 22A22, 70F25, 70G45, 70G65, 70H03, 70H05,70Q05, 70S05.

Key words and phrases. Lie algebroids, Lie groupoids, Lagrangian Mechanics, HamiltonianMechanics, nonholonomic Lagrangian systems, mechanical control systems, Discrete Mechanics,Classical Field Theory.

This work has been partially supported by MEC (Spain) Grants MTM 2004-7832, BFM2003-01319 and BFM2003-02532.

1

2 J. CORTES, M. DE LEON, J. C. MARRERO, D. MARTIN DE DIEGO, AND E. MARTINEZ

6.1. Lie algebroid structure on the vector bundle πτ : EG

G× EG −→ G 296.2. Discrete Variational Mechanics on Lie groupoids 307. Classical Field Theory on Lie algebroids 337.1. Jets 337.2. Morphisms and admissible maps 357.3. Variational Calculus 357.4. Examples 378. Future work 38References 39

1. Introduction

The theory of Lie algebroids and Lie groupoids has proved to be very useful indifferent areas of mathematics including algebraic and differential geometry, alge-braic topology, and symmetry analysis. In this survey, we illustrate the wide rangeof applications of this formalism to Mechanics. Specifically, we show how the flexi-bility provided by Lie algebroids and groupoids allows us to analyze, within a singleframework, different classes of situations such as systems subject to nonholonomicconstraints, mechanical control systems, Discrete Mechanics and Field Theory.

The notions of Lie algebroid and Lie groupoid allow to study general Lagrangianand Hamiltonian systems beyond the ones defined on the tangent and cotangentbundles of the configuration manifold, respectively. These include systems deter-mined by Lagrangian and Hamiltonian functions defined on Lie algebras, Lie groups,Cartesian products of manifolds, and reduced spaces.

The inclusive feature of the Lie algebroid formalism is particularly relevant forthe class of Lagrangian systems invariant under the action of a Lie group of symme-tries. Given a standard Lagrangian system, one associates to the Lagrangian func-tion a Poincare-Cartan symplectic form and an energy function using the particulargeometry of the tangent bundle. The dynamics is then obtained as the Hamiltonianvector field associated to the energy function through the Poincare-Cartan form.The reduction by the Lie group action of the dynamics of this system yields areduced dynamics evolving on a quotient space (which is not a tangent bundle).However, the interplay between the geometry of this quotient space and the reduceddynamics is not as transparent as in the tangent bundle case. Recent efforts havelead to a unifying geometric framework to overcome this drawback. It is preciselythe underlying structure of Lie algebroid on the phase space what allows a unifiedtreatment. This idea was introduced by Weinstein [54] (see also [31]), who devel-oped a generalized theory of Lagrangian Mechanics on Lie algebroids. He obtainedthe equations of motion using the linear Poisson structure on the dual of the Liealgebroid and the Legendre transformation associated with a (regular) Lagrangian.In [54], Weinstein also posed the question of whether it was possible to develop atreatment on Lie algebroids and groupoids similar to Klein’s formalism for ordinaryLagrangian Mechanics [22]. This question was answered positively by E. Martınezin [38] (see also [13, 39, 41, 49]). The main notion was that of prolongation of a Liealgebroid over a mapping, introduced by P.J. Higgins and K. Mackenzie [19]. Morerecently, the work [26] has developed a description of Hamiltonian and Lagrangiandynamics on a Lie algebroid in terms of Lagrangian submanifolds of symplectic Lie

A SURVEY OF MECHANICS ON LIE ALGEBROIDS AND GROUPOIDS 3

algebroids. An alternative approach, using the linear Poisson structure on the dualof the Lie algebroid was discussed in [18].

In this paper, we present an overview of these developments. We make specialemphasis on one of the main advantages of the Lie algebroid formalism: the pos-sibility of establishing appropriate maps (called morphisms) between systems thatrespect the structure of the phase space, and allow to relate their respective prop-erties. As we will show, this will allow us to present a comprehensive study of thereduction process of Lagrangian systems while remaining within the same categoryof mathematical objects.

We also consider nonholonomic systems (i.e., systems subject to constraints in-volving the velocities, see [10] for a list of references) and control systems evolvingon Lie algebroids [13]. This is motivated by the renewed interest in the study ofnonholonomic mechanical systems for new applications in the areas of robotics andcontrol. In particular, we provide widely applicable tests to decide the accessi-bility and controllability properties of mechanical control systems defined on Liealgebroids.

We end this survey by paying attention to two recent developments in the contextof Lie algebroids and groupoids: Discrete Mechanics and Classical Field Theory.Discrete Mechanics seeks to develop a complete discrete-time counterpart of theusual continuous-time treatment of Mechanics. The ultimate objective of this effortis the construction of numerical integrators for Lagrangian and Hamiltonian systems(see [37] and references therein). Up to now, this effort has been mainly focusedon the case of discrete Lagrangian functions defined on the Cartesian product ofthe configuration manifold with itself. This Cartesian product is just an exampleof a Lie groupoid. Here, we review the recent developments in [35], where weproposed a complete description of Lagrangian and Hamiltonian Mechanics on Liegroupoids. In particular, this description covers the analysis of discrete systemswith symmetries, and naturally produces reduced geometric integrators. Anotherextension that we consider is the study of Classical Field Theory on Lie algebroids.Thinking of a Lie algebroid as a substitute of the tangent bundle of a manifold, wesubstitute the classical notion of fibration bundle by a surjective morphism of Liealgebroids π : E −→ F . Then, we construct the jet space Jπ as the affine bundlewhose elements are linear maps from a fiber of F to a fiber of E, i.e., sections ofthe projection π. After a suitable choice of the space of variations, we derive theEuler-Lagrange equations for this problem.

The paper is organized as follows. In Section 2 we present some basic facts onLie algebroids, including results from differential calculus, morphisms and prolon-gations of Lie algebroids, and linear connections. We also introduce the notionof Lie groupoid, Lie algebroid associated to a Lie groupoid, and morphisms andprolongations of Lie groupoids. Various examples are given in order to illustratethe generality of the theory. In Section 3, we give a brief introduction to the La-grangian formalism of Mechanics on Lie algebroids, determined by a Lagrangianfunction L : E −→ R on the Lie algebroid τ : E −→ M . Likewise, we introduce theHamiltonian formalism on Lie algebroids, determined by a Hamiltonian functionH : E∗ −→ R, where τ∗ : E∗ −→ M is the dual of the Lie algebroid E −→ M . InSection 4 we introduce the class of nonholonomic Lagrangian systems. We studythe existence and uniqueness of solutions, and characterize the notion of regularityof a nonholonomic system. Under this property, we derive a procedure to obtainthe solution of the nonholonomic problem from the solution of the free problem bymeans of projection techniques. Moreover, we construct a nonholonomic bracketthat measures the evolution of the observables, and we study the reduction of


nonholonomic systems in terms of morphisms of Lie algebroids. In Section 5 weintroduce the class of mechanical control systems defined on a Lie algebroid. Wegeneralize the notion of affine connection control system to the setting of Lie alge-broids, and introduce the notions of (base) accessibility and (base) controllability.We provide sufficient conditions to check these properties for a given mechanicalcontrol system. In Section 6, we study Discrete Mechanics on Lie groupoids. Inparticular, we construct the discrete Euler-Lagrange equations, discrete Poincare-Cartan sections, discrete Legendre transformations, and Noether’s theorem, andidentify the preservation properties of the discrete flow. In the last section, weextend the variational formalism for Classical Field Theory to the setting of Liealgebroids. Given a Lagrangian function, we study the problem of finding criti-cal points of the action functional when we restrict the fields to be morphisms ofLie algebroids. Throughout the paper, various examples illustrate the results. Weconclude the paper by identifying future directions of research.

2. Lie algebroids and Lie groupoids

2.1. Lie algebroids. Given a real vector bundle τ : E −→ M , let Sec(τ) denotethe space of the global cross sections of τ : E −→ M . A Lie algebroid E over amanifold M is a real vector bundle τ : E −→ M together with a Lie bracket [[·, ·]]on Sec(τ) and a bundle map ρ : E −→ TM over the identity, called the anchormap, such that the homomorphism (denoted also ρ : Sec(τ) −→ X(M)) of C∞(M)-modules induced by the anchor map verifies

[[X, fY ]] = f [[X, Y ]] + ρ(X)(f)Y,

for X,Y ∈ Sec(τ) and f ∈ C∞(M). The triple (E, [[·, ·]], ρ) is called a Lie algebroidover M (see [32, 33]). If (E, [[·, ·]], ρ) is a Lie algebroid over M, then the anchor mapρ : Sec(τ) −→ X(M) is a homomorphism between the Lie algebras (Sec(τ), [[·, ·]])and (X(M), [·, ·]).

In what concerns to Mechanics, it is convenient to think of a Lie algebroid asa generalization of the tangent bundle of M . One regards an element a of E asa generalized velocity, and the actual velocity v is obtained when applying theanchor to a, i.e., v = ρ(a). A curve a : [t0, t1] −→ E is said to be admissible ifm(t) = ρ(a(t)), where m(t) = τ(a(t)) is the base curve.

Given local coordinates (xi) in the base manifold M and a local basis of sections(eα) of E, then local coordinates of a point a ∈ E are (xi, yα) where a = yαeα(τ(a)).In local form, the Lie algebroid structure is determined by the local functions ρi

α

and Cγαβ on M . Both are determined by the relations

ρ(eα) = ρiα

∂

∂xi, (2.1)

[[eα, eβ ]] = Cγαβ eγ (2.2)

and they satisfy the following equations

ρjα

∂ρiβ

∂xj− ρj

β

∂ρiα

∂xj= ρi

γCγαβ and

∑

cyclic(α,β,γ)

[ρi

α

∂Cνβγ

∂xi+ Cµ

βγCναµ

]= 0. (2.3)


Cartan calculus. One may define the exterior differential of E, d : Sec(∧kτ∗) −→Sec(∧k+1τ∗), as follows

dω(X0, . . . , Xk) =k∑

i=0

(−1)iρ(Xi)(ω(X0, . . . , Xi, . . . , Xk))

+∑

i<j

(−1)i+jω([[Xi, Xj ]], X0, . . . , Xi, . . . , Xj , . . . , Xk),(2.4)

for ω ∈ Sec(∧kτ∗) and X0, . . . , Xk ∈ Sec(τ). d is a cohomology operator, that is,d2 = 0. In particular, if f : M −→ R is a real smooth function then df(X) = ρ(X)f,for X ∈ Sec(τ). Locally,

dxi = ρiαeα and deγ = −1

2Cγ

αβeα ∧ eβ ,

where eα is the dual basis of eα. We may also define the Lie derivative withrespect to a section X of E as the operator LX : Sec(∧kτ∗) −→ Sec(∧kτ∗) givenby LX = iX d + d iX (for more details, see [32, 33]).Morphisms. Let (E, [[·, ·]], ρ) (resp., (E′, [[·, ·]]′, ρ′)) be a Lie algebroid over a man-ifold M (resp., M ′) and suppose that Ψ: E −→ E′ is a vector bundle morphismover the map Ψ0 : M −→ M ′. Then, the pair (Ψ, Ψ0) is said to be a Lie algebroidmorphism if

d((Ψ, Ψ0)∗φ′) = (Ψ, Ψ0)∗(d′φ′), for all φ′ ∈ Sec(∧k(E′)∗) and for all k, (2.5)

where d (resp., d′) is the differential of the Lie algebroid E (resp., E′) (see [26]).Note that (Ψ,Ψ0)∗φ′ is the section of the vector bundle ∧kE∗ −→ M defined fork > 0 by

((Ψ,Ψ0)∗φ′)x(a1, . . . , ak) = φ′Ψ0(x)(Ψ(a1), . . . , Ψ(ak)),

for x ∈ M and a1, . . . , ak ∈ Ex, and by (Ψ,Ψ0)∗f = f Ψ0 for f ∈ Sec(∧0E′∗) =C∞(M ′). In the particular case when M = M ′ and Ψ0 = idM then (2.5) holds ifand only if

[[Ψ X, Ψ Y ]]′ = Ψ[[X, Y ]], ρ′(ΨX) = ρ(X), for X, Y ∈ Sec(τ).

Linear connections on Lie algebroids. Let τ : E −→ M be a Lie algebroid overM . A connection on E is a R-bilinear map ∇ : Sec(E)×Sec(E) −→ Sec(E) suchthat

∇fXY = f∇XY , ∇X(fY ) = ρ(X)(f)Y + f∇XY

for f ∈ C∞(M) and X,Y ∈ Sec(E).Given a local basis eα of Sec(E) such that X = Xαeα and Y = Y βeβ then

∇XY = Xα

(ρi

α

∂Y γ

∂xi+ Γγ

αβY β

)eγ .

The terms Γγαβ are called the connection coefficients. The symmetric product

associated with ∇ is given by

〈X : Y 〉 = ∇XY +∇Y X, X, Y ∈ Sec(E).

Since the connection is C∞(M)-linear in the first argument, it is possible todefine the derivative of a section Y ∈ Sec(E) with respect to an element a ∈ Em

by simply putting∇aY = (∇XY )(m),

with X ∈ Sec(E) satisfying X(m) = a. Moreover, the connection allows us totake the derivative of sections along maps and, as a particular case, of sectionsalong curves. If we have a morphism of Lie algebroids Φ : F −→ E over the map


ϕ : N −→ M and a section X : N −→ E along ϕ (i.e., X(n) ∈ Eϕ(n), for n ∈ N),then X may be written as

X =p∑

l=1

Fl(Xl ϕ),

for some sections X1, . . . , Xp of E and for some functions F1, . . . , Fp ∈ C∞(N),and the derivative of X along ϕ is given by

∇bX =p∑

l=1

[(ρF (b)Fl)Xl(ϕ(n)) + Fl(n)∇Φ(b)Xl

], for b ∈ Fn ,

where ρF is the anchor map of the Lie algebroid F −→ N .A particular case of the above general situation is the following. Let a : I −→ E

be an admissible curve and b : I −→ E be a curve in E, both of them projecting byτ onto the same base curve in M , τ(a(t)) = m(t) = τ(b(t)). Take the Lie algebroidstructure TI −→ I and consider the morphism Φ : TI −→ E, Φ(t, t) = ta(t) overm : I −→ M . Then one can define the derivative of b(t) along a(t) as ∇d/dtb(t).This derivative is usually denoted by ∇a(t)b(t). In local coordinates, this reads

∇a(t)b(t) =[dbγ

dt+ Γγ

αβaαbβ

]eγ(m(t)), for all t.

The admissible curve a : I −→ E is said to be a geodesic for ∇ if ∇a(t)a(t) = 0(see [13]).

Now, let G : E ×M E −→ R be a bundle metric on a Lie algebroid τ : E −→ M .In a parallel way to the situation in the tangent bundle geometry, one can see thatthere is a canonical connection ∇G on E associated with G. In fact, the connection∇G is determined by the formula

2G(∇GXY,Z) = ρ(X)(G(Y,Z)) + ρ(Y )(G(X, Z))− ρ(Z)(G(X, Y ))

+G(X, [[Z, Y ]]) + G(Y, [[Z, X]])− G(Z, [[Y, X]]),

for X, Y, Z ∈ Sec(E). ∇G is a torsion-less connection and it is metric with respectto G. In other words

[[X,Y ]] = ∇GXY −∇G

Y X ,

ρ(X)(G(Y, Z)) = G(∇GXY, Z) + G(Y,∇G

XZ) .

∇G is called the Levi-Civita connection of G (see [13]).Finally, suppose that E = D ⊕ Dc, with D and Dc vector subbundles of E,

and denote by P : E −→ D and Q : E −→ Dc the corresponding complementaryprojectors induced by the decomposition. Then, the constrained connection isthe connection ∇ on E defined by

∇XY = P (∇XY ) +∇X(QY ),

for X,Y ∈ Sec(E) (for the properties of the constrained connection ∇, see [13]).Examples. We will present some examples of Lie algebroids.

1.- Real Lie algebras of finite dimension. Let g be a real Lie algebra offinite dimension. Then, it is clear that g is a Lie algebroid over a single point.

2.- The tangent bundle. Let TM be the tangent bundle of a manifold M .Then, the triple (TM, [·, ·], idTM ) is a Lie algebroid over M , where idTM : TM −→TM is the identity map.

3.- Foliations. Let F be a foliation of finite dimension on a manifold P andτF : TF −→ P be the tangent bundle to the foliation F. Then, τF : TF −→ P is aLie algebroid over P . The anchor map is the canonical inclusion ρF : TF −→ TP


and the Lie bracket on the space Sec(τF) is the restriction to Sec(τF) of the standardLie bracket of vector fields on P . In particular, if π : P −→ M is a fibration,τP : TP −→ P is the canonical projection and (τP )|V π : V π −→ P is the restrictionof τP to the vertical bundle to π, then (τP )|V π : V π −→ P is a Lie algebroid over P .

4.- Atiyah algebroids. Let p : Q −→ M be a principal G-bundle. Denote byΦ : G × Q −→ Q the free action of G on Q and by TΦ : G × TQ −→ TQ thetangent action of G on TQ. Then, one may consider the quotient vector bundleτQ|G : TQ/G −→ M = Q/G, and the sections of this vector bundle may beidentified with the vector fields on Q which are invariant under the action Φ. Usingthat every G-invariant vector field on Q is p-projectable and that the usual Liebracket on vector fields is closed with respect to G-invariant vector fields, we caninduce a Lie algebroid structure on TQ/G. This Lie algebroid is called the Atiyahalgebroid associated with the principal G-bundle p : Q −→ M (see [26, 32]).

5.- Action Lie algebroids. Let (E, [[·, ·]], ρ) be a Lie algebroid over a manifoldM and f : M ′ −→ M be a smooth map. Then, the pull-back of E over f , f∗E = (x′, a) ∈ M ′ × E | f(x′) = τ(a) , is a vector bundle over M ′ whose vector bundleprojection is the restriction to f∗E of the first canonical projection pr1 : M ′×E →M ′. However, f∗E is not, in general, a Lie algebroid.

Now, suppose that Φ: Sec(τ) −→ X(M ′) is an action of E on f , that is, Φ is aR-linear map which satisfies the following conditions

Φ(hX) = (h f)ΦX, Φ[[X, Y ]] = [ΦX, ΦY ], ΦX(h f) = ρ(X)(h) f,

for X, Y ∈ Sec(τ) and h ∈ C∞(M). Then, one may introduce a Lie algebroidstructure ([[·, ·]]Φ, ρΦ) on the vector bundle f∗E → M ′ which is characterized by thefollowing conditions

[[X f, Y f ]]Φ = [[X, Y ]] f, ρΦ(X f) = Φ(X), for X, Y ∈ Sec(τ). (2.6)

The resultant Lie algebroid is denoted by E n f and we call it an action Liealgebroid (for more details, see [26]).

6.- The prolongation of a Lie algebroid over a fibration [19, 26, 39]. Let(E, [[·, ·]], ρ) be a Lie algebroid over a manifold M and π : P −→ M be a fibration.We consider the subset of E × TP

T Ep P = (b, v) ∈ Ex × TpP | ρ(b) = Tpπ(v) ,

where Tπ : TP −→ TM is the tangent map to π, p ∈ Px and π(p) = x . We willfrequently use the redundant notation (p, b, v) to denote the element (b, v) ∈ T E

p P .T EP = ∪p∈PT E

p P is a vector bundle over P and the vector bundle projection τEP

is just the projection onto the first factor. The anchor of T EP is the projectiononto the third factor, that is, the map ρπ : T EP −→ TP given by ρπ(p, b, v) = v.The projection onto the second factor will be denoted by T π : T EP −→ E, and itis a morphism of Lie algebroids over π. Explicitly, T π(p, b, v) = b.

An element z ∈ T EP is said to be vertical if it projects to zero, that is T π(z) =0. Therefore it is of the form (p, 0, v), with v a π-vertical vector tangent to P at p.

Given local coordinates (xi, uA) on P and a local basis eα of sections of E, wecan define a local basis Xα, VA of sections of T EP by

Xα(p) =(p, eα(π(p)), ρi

α

∂

∂xi

∣∣∣p

)and VA(p) =

(p, 0,

∂

∂uA

∣∣∣p

).

If z = (p, b, v) is an element of T EP , with b = zαeα, then v is of the form v =ρi

αzα ∂∂xi + vA ∂

∂uA , and we can write

z = zαXα(p) + vAVA(p).


Vertical elements are linear combinations of VA.The anchor map ρπ applied to a section Z of T EP with local expression Z =

ZαXα + V AVA is the vector field on P whose coordinate expression is

ρπ(Z) = ρiαZα ∂

∂xi+ V A ∂

∂uA.

Next, we will see that it is possible to induce a Lie bracket structure on thespace of sections of T EP . For that, we say that a section X of τE

P : T EP −→ Pis projectable if there exists a section X of τ : E −→ M and a vector fieldU ∈ X(P ) which is π-projectable to the vector field ρ(X) and such that X(p) =(X(π(p)), U(p)), for all p ∈ P . For such a projectable section X, we will use thefollowing notation X ≡ (X,U). It is easy to prove that one may choose a localbasis of projectable sections of the space Sec(τE

P ).The Lie bracket of two projectable sections Z1 = (X1, U1) and Z2 = (X2, U2) is

then given by

[[Z1, Z2]]π(p) = (p, [[X1, X2]](x), [U1, U2](p)), p ∈ P, x = π(p).

Since any section of T EP can be locally written as a linear combination of pro-jectable sections, the definition of the Lie bracket for arbitrary sections of T EPfollows. In particular, the Lie brackets of the elements of the basis are

[[Xα,Xβ ]]π = Cγαβ Xγ , [[Xα, VB ]]π = 0 and [[VA,VB ]]π = 0,

and, therefore, the exterior differential is determined by

dxi = ρiαXα, duA = VA,

dXγ = −12Cγ

αβXα ∧ Xβ , dVA = 0,

where Xα, VA is the dual basis to Xα, VA.The Lie algebroid T EP is called the prolongation of E over π or the E-

tangent bundle to π.

2.2. Lie groupoids. In this section, we review the definition of a Lie groupoid andpresent some basic facts generalities about them (see [32, 33] for more details). Agroupoid over a set M is a set G together with the following structural maps:

• A pair of maps α : G −→ M , the source, and β : G −→ M , the target.These maps define the set of composable pairs

G2 = (g, h) ∈ G×G | β(g) = α(h) .

• A multiplication m : G2 −→ G, to be denoted simply by m(g, h) = gh,such that

– α(gh) = α(g) and β(gh) = β(h),– g(hk) = (gh)k.

• An identity section ε : M −→ G such that– ε(α(g))g = g and gε(β(g)) = g.

• An inversion map i : G −→ G, to be denoted simply by i(g) = g−1,such that

– g−1g = ε(β(g)) and gg−1 = ε(α(g)).

A groupoid G over a set M will be denoted simply by the symbol G ⇒ M .The groupoid G ⇒ M is said to be a Lie groupoid if G and M are manifolds and

all the structural maps are differentiable with α and β differentiable submersions.If G ⇒ M is a Lie groupoid then m is a submersion, ε is an immersion and i is


a diffeomorphism. Moreover, if x ∈ M , α−1(x) (resp., β−1(x)) will be said theα-fiber (resp., the β-fiber) of x.

On the other hand, if g ∈ G then the left-translation by g ∈ G and theright-translation by g are the diffeomorphisms

lg : α−1(β(g)) −→ α−1(α(g)) ; h −→ lg(h) = gh,rg : β−1(α(g)) −→ β−1(β(g)) ; h −→ rg(h) = hg.

Note that l−1g = lg−1 and r−1

g = rg−1 .

A vector field X on G is said to be left-invariant (resp., right-invariant) ifit is tangent to the fibers of α (resp., β) and X(gh) = (Thlg)(Xh) (resp., X(gh) =(Tgrh)(X(g)), for (g, h) ∈ G2.

Now, we will recall the definition of the Lie algebroid associated with G.We consider the vector bundle τ : EG −→ M , whose fiber at a point x ∈ M

is (EG)x = Vε(x)α = Ker(Tε(x)α). It is easy to prove that there exists a bijectionbetween the space Sec(τ) and the set of left-invariant (resp., right-invariant) vectorfields on G. If X is a section of τ : EG −→ M , the corresponding left-invariant(resp., right-invariant) vector field on G will be denoted

←−X (resp.,

−→X ), where

←−X (g) = (Tε(β(g))lg)(X(β(g))), (2.7)

−→X (g) = −(Tε(α(g))rg)((Tε(α(g))i)(X(α(g)))), (2.8)

for g ∈ G. Using the above facts, we may introduce a Lie algebroid structure([[·, ·]], ρ) on EG, which is defined by

←−−−−[[X,Y ]] = [

←−X,←−Y ], ρ(X)(x) = (Tε(x)β)(X(x)), (2.9)

for X,Y ∈ Sec(τ) and x ∈ M (for more details, see [9, 32]).Given two Lie groupoids G ⇒ M and G′ ⇒ M ′, a morphism of Lie groupoids

is a smooth map Ψ : G −→ G′ such that

(g, h) ∈ G2 =⇒ (Ψ(g),Ψ(h)) ∈ (G′)2

and

Ψ(gh) = Ψ(g)Ψ(h).

A morphism of Lie groupoids Ψ : G −→ G′ induces a smooth map Φ0 : M −→ M ′

in such a way that

α′ Ψ = Φ0 α, β′ Ψ = Φ0 β, Ψ ε = ε′ Φ0,

α, β and ε (resp., α′, β′ and ε′) being the source, the target and the identity sectionof G (resp., G′).

Suppose that (Ψ, Φ0) is a morphism between the Lie groupoids G ⇒ M andG′ ⇒ M ′ and that τ : EG −→ M (resp., τ ′ : EG′ −→ M ′) is the Lie algebroid ofG (resp., G′). Then, if x ∈ M we may consider the linear map Φx : (EG)x −→(EG′)Φ0(x) defined by

Φx(vε(x)) = (Tε(x)Ψ)(vε(x)), for vε(x) ∈ AxG. (2.10)

In fact, we have that the pair (Φ,Φ0) is a morphism between the Lie algebroidsτ : EG −→ M and τ ′ : EG′ −→ M ′ (see [32, 33]).


Examples. We will present some examples of Lie groupoids.1.- Lie groups. Any Lie group G is a Lie groupoid over e, the identity element

of G. The Lie algebroid associated with G is just the Lie algebra g of G.2.- The pair or banal groupoid. Let M be a manifold. The product manifold

M × M is a Lie groupoid over M in the following way: α is the projection ontothe first factor and β is the projection onto the second factor; ε(x) = (x, x), for allx ∈ M , m((x, y), (y, z)) = (x, z), for (x, y), (y, z) ∈ M × M and i(x, y) = (y, x).M ×M ⇒ M is called the pair or banal groupoid. If x is a point of M , it followsthat

Vε(x)α = 0x × TxM ⊆ TxM × TxM ∼= T(x,x)(M ×M).

Thus, the linear maps

Φx : TxM −→ Vε(x)α, vx −→ (0x, vx),

induce an isomorphism (over the identity of M) between the Lie algebroids τM :TM −→ M and τ : EM×M −→ M.

3.- The Lie groupoid associated with a fibration. Let π : P −→ M bea fibration, that is, π is a surjective submersion and denote by Gπ the subset ofP × P given by

Gπ = (p, p′) ∈ P × P | π(p) = π(p′) .

Then, Gπ is a Lie groupoid over P and the structural maps απ, βπ, mπ, επ and iπare the restrictions to Gπ of the structural maps of the pair groupoid P × P ⇒ P .

If p is a point of P it follows that

Vεπ(p)απ = (0p, Yp) ∈ TpP × TpP | (Tpπ)(Yp) = 0 .

Thus, if (τP )|V π : V π −→ P is the vertical bundle to π then the linear maps

(Φπ)p : Vpπ −→ Vεπ(p)απ, Yp −→ (0p, Yp)

induce an isomorphism (over the identity of M) between the Lie algebroids (τP )|V π :V π −→ P and τ : EGπ −→ P .

4.- Atiyah or gauge groupoids. Let p : Q −→ M be a principal left G-bundle.Then, the free action Φ : G×Q −→ Q, (g, q) −→ Φ(g, q) = gq, of G on Q induces,in a natural way, a free action Φ×Φ : G× (Q×Q) −→ Q×Q of G on Q×Q givenby (Φ×Φ)(g, (q, q′)) = (gq, gq′), for g ∈ G and (q, q′) ∈ Q×Q. Moreover, one mayconsider the quotient manifold (Q × Q)/G which admits a Lie groupoid structureover M with structural maps given by

α : (Q×Q)/G −→ M ; [(q, q′)] −→ p(q),β : (Q×Q)/G −→ M ; [(q, q′)] −→ p(q′),ε : M −→ (Q×Q)/G ; x −→ [(q, q)], if p(q) = x,m : ((Q×Q)/G)2 −→ (Q×Q)/G ; ([(q, q′)], [(gq′, q′′)]) −→ [(gq, q′′)],i : (Q×Q)/G −→ (Q×Q)/G ; [(q, q′)] −→ [(q′, q)].

This Lie groupoid is called the Atiyah (gauge) groupoid associated with theprincipal G-bundle p : Q −→ M (see [31]).

If x is a point of M such that p(q) = x, with q ∈ Q, and pQ×Q : Q × Q −→(Q×Q)/G is the canonical projection then it is clear that

Vε(x)α = (T(q,q)pQ×Q)(0q × TqQ).

Thus, if τQ|G : TQ/G −→ M is the Atiyah algebroid associated with the principalG-bundle p : G −→ M then the linear maps

(TQ/G)x −→ Vε(x)α ; [vq] −→ (T(q,q)pQ×Q)(0q, vq), with vq ∈ TqQ,


induce an isomorphism (over the identity of M) between the Lie algebroids τ :E(Q×Q)/G −→ M and τQ|G : TQ/G −→ M .

5.- Action Lie groupoids. Let G ⇒ M be a Lie groupoid and f : M ′ −→ Mbe a smooth map. If M ′

f×α G = (p, g) ∈ P ×G | f(p) = α(g) then a rightaction of G on f is a smooth map

M ′f×α G −→ M ′, (x′, g) −→ x′g,

which satisfies the following relations

f(x′g) = β(g), for (x′, g) ∈ M ′f×α G,

(x′g)h = x′(gh), for (x′, g) ∈ M ′f×α G and (g, h) ∈ G2, and

x′ε(f(x′)) = x′, for x′ ∈ M ′.

Given such an action one constructs the action Lie groupoid M ′f×α G over

M ′ by defining

αf : M ′f×α G −→ M ′ ; (x′, g) −→ x′,

βf : M ′f×α G −→ M ′ ; (x′, g) −→ x′g,

εf : M ′ −→ M ′f×α G ; x′ −→ (x′, ε(f(x′))),

mf : (M ′f×α G)2 −→ M ′

f×α G ; ((x′, g), (x′g, h)) −→ (x′, gh),if : M ′

f×α G −→ M ′f×α G ; (x′, g) −→ (x′g, g−1).

Now, if x′ ∈ M ′, we consider the map x′ · : α−1(f(x′)) −→ M ′ given by

x′ · (g) = x′g.

Then, if τ : EG −→ M is the Lie algebroid of G, the R-linear map Φ: Sec(τ) −→X(M ′) defined by

Φ(X)(x′) = (Tε(f(x′))x′ ·)(X(f(x′))), for X ∈ Sec(τ) and x′ ∈ M ′,

induces an action of EG on f : M ′ −→ M . In addition, the Lie algebroid associatedwith the Lie groupoid M ′

f×α G ⇒ P is the action Lie algebroid EG n f (for moredetails, see [19]).

6.- The prolongation of a Lie groupoid over a fibration. Given a Liegroupoid G ⇒ M and a fibration π : P −→ M , we consider the set

PG×P ≡ P π×α G β×π P = (p, g, p′) ∈ P ×G× P | π(p) = α(g), β(g) = π(p′) .

Then, PG× P is a Lie groupoid over P with structural maps given by

απ : PG× P −→ P ; (p, g, p′) −→ p,

βπ : PG× P −→ P ; (p, g, p′) −→ p′,

επ : P −→ PG× P ; p −→ (p, ε(π(p)), p),

mπ : (PG× P )2 −→ P

G× P ; ((p, g, p′), (p′, h, p′′)) −→ (p, gh, p′′),

iπ : PG× P −→ P

G× P ; (p, g, p′) −→ (p′, g−1, p).

PG× P is called the prolongation of G over π : P −→ M .Now, denote by τ : EG −→ M the Lie algebroid of G, by E

PG×P

the Lie algebroid

of PG× P and by T EGP the prolongation of τ : EG −→ M over the fibration π. If

p ∈ P and m = π(p), then it follows that(

EP

G×P

)

p

=

(0p, vε(m), Xp) ∈ TpP × (EG)m × TpP∣∣ (Tpπ)(Xp) = (Tε(m)β)(vε(m))


and, thus, one may consider the linear isomorphism

(Φπ)p : (EP

G×P)p −→ T EG

p P, (0p, vε(m), Xp) −→ (vε(m), Xp). (2.11)

In addition, one may prove that the maps (Φπ)p, p ∈ P , induce an isomorphismΦπ : E

PG×P

−→ T EGP between the Lie algebroids EP

G×Pand T EGP (for more

details, see [19]).A particular case. Next, suppose that P = EG and that the map π is just the

vector bundle projection τ : EG −→ M . In this case,

EG

G× EG = EG τ×α G β×τ EG

and we may define the map Θ : EG

G× EG −→ V β ⊕G V α as follows

Θ(uε(α(g)), g, vε(β(g))) = ((Tε(α(g))(rg i))(uε(α(g))), (Tε(β(g))lg)(vε(β(g)))),

for (uε(α(g)), g, vε(β(g))) ∈ (EG)α(g) ×G× (EG)β(g). Θ is a bijective map and

Θ−1(Xg, Yg) = ((Tg(i rg−1))(Xg), g, (Tglg−1)(Yg)),

for (Xg, Yg) ∈ Vgβ ⊕ Vgα. Thus, the spaces EG

G× EG and V β ⊕G V α may beidentified and, under this identification, the structural maps of the Lie groupoidstructure on V β ⊕G V α are given by

ατ : V β ⊕G V α −→ EG ; (Xg, Yg) −→ (Tg(i rg−1))(Xg),βτ : V β ⊕G V α −→ EG ; (Xg, Yg) −→ (Tglg−1)(Yg),ετ : EG −→ V β ⊕G V α ; vε(x) −→ ((Tε(x)i)(vε(x)), vε(x)),iτ : V β ⊕G V α −→ V β ⊕G V α ; (Xg, Yg) −→ ((Tgi)(Yg), (Tgi)(Xg)),

and the multiplication mτ : (V β ⊕G V α)2 −→ V β ⊕G V α is

mτ ((Xg, Yg), ((Tg(rgh i))(Yg), Zh)) = ((Tgrh)(Xg), (Thlg)(Zh)).

This Lie groupoid structure was considered by Saunders [53]. We remark that the

Lie algebroid of EG

G× EG∼= V β ⊕G V α ⇒ EG is isomorphic to the prolongation

T EGEG of EG over τ : EG −→ M .

3. Mechanics on Lie algebroids

We recall that a symplectic section on a vector bundle π : F −→ M is a sectionω of ∧2π∗ which is regular at every point when it is considered as a bilinear form.By a symplectic Lie algebroid we mean a pair (E, ω) where τ : E −→ M is aLie algebroid and ω is a symplectic section on the vector bundle E satisfying thecompatibility condition dω = 0, where d is the exterior differential of E.

On a symplectic Lie algebroid (E, ω) we can define a dynamical system for everyfunction on the base, as in the standard case of a tangent bundle. Given a functionH ∈ C∞(M) there is a unique section σH ∈ Sec(τ) such that

iσHω = dH.

The section σH is said to be the Hamiltonian section defined by H and thevector field XH = ρ(σH) is said to be the Hamiltonian vector field defined byH. In this way we get the dynamical system x = XH(x).

A symplectic structure ω on a Lie algebroid E defines a Poisson bracket , ω

on the base manifold M as follows. Given two functions F, G ∈ C∞(M) we definethe bracket

F,Gω = ω(σF , σG).


It is easy to see that the closure condition dω = 0 implies that , ω is a Poissonstructure on M . In other words, if we denote by Λ the inverse of ω as bilinear form,then F, Gω = Λ(dF, dG). The Hamiltonian dynamical system associated to Hcan be written in terms of the Poisson bracket as x = x,Hω.

By a symplectomorphism between two symplectic Lie algebroids (E, ω) and(E′, ω′) we mean an isomorphism of Lie algebroids (Ψ, Ψ0) from E to E′ such that(Ψ,Ψ0)∗ω′ = ω. In this case the base map Ψ0 is a Poisson diffeomorphism, that is,it satisfies Ψ∗0F ′, G′ω′ = Ψ∗0F ′, Ψ∗0G′ω, for all F ′, G′ ∈ C∞(M ′).

Sections 3.1 and 3.2 describe two particular and important cases of the aboveconstruction.

3.1. Lagrangian Mechanics. In [38] (see also [26]) a geometric formalism forLagrangian Mechanics on Lie algebroids was introduced. It is developed in theprolongation T EE of a Lie algebroid E over the vector bundle projection τ : E −→M . The canonical geometrical structures defined on T EE are the following:

• The vertical lift ξV : τ∗E −→ T EE given by ξV (a, b) = (a, 0, bVa ), where

bVa is the vector tangent to the curve a + tb at t = 0.

• The vertical endomorphism S : T EE −→ T EE defined as follows:

S(a, b, v) = ξV (a, b) = (a, 0, bV

a ).

• The Liouville section, which is the vertical section corresponding to theLiouville dilation vector field:

∆(a) = ξV (a, a) = (a, 0, aV

a ).

We also mention that the complete lift XC of a section X ∈ Sec(E) is thesection of T EE characterized by the following properties:

(i) projects to X, i.e., T τ XC = X τ ,(ii) LXC µ = LXµ,

where by α ∈ C∞(E) we denote the linear function associated to α ∈ Sec(E∗).Given a Lagrangian function L ∈ C∞(E) we define the Cartan 1-section

θL ∈ Sec((T EE)∗) and the Cartan 2-section ωL ∈ Sec(∧2(T EE)∗) and theLagrangian energy EL ∈ C∞(E) as

θL = S∗(dL), ωL = −dθL and EL = L∆L− L. (3.1)

If (xi, yα) are local fibred coordinates on E, (ρiα, Cγ

αβ) are the corresponding localstructure functions on E and Xα, Vα is the corresponding local basis of sectionsof T EE then

SXα = Vα, SVα = 0, for all α, (3.2)

∆ = yαVα, (3.3)

ωL =∂2L

∂yα∂yβXα ∧ Vβ +

12

(∂2L

∂xi∂yαρi

β −∂2L

∂xi∂yβρi

α +∂L

∂yγCγ

αβ

)Xα ∧ Xβ , (3.4)

EL =∂L

∂yαyα − L. (3.5)

From (3.2), (3.3), (3.4) and (3.5), it follows that

iSXωL = −S∗(iXωL), i∆ωL = −S∗(dEL), (3.6)

for X ∈ Sec(T EE).


Now, a curve t −→ c(t) on E is a solution of the Euler-Lagrange equationsfor L if

- c is admissible (that is, ρ(c(t)) = m(t), where m = τ c) and- i(c(t),c(t))ωL(c(t))− dEL(c(t)) = 0, for all t.

If c(t) = (xi(t), yα(t)) then c is a solution of the Euler-Lagrange equations for L ifand only if

xi = ρiαyα,

d

dt

( ∂L

∂yα

)+

∂L

∂yγCγ

αβyβ − ρiα

∂L

∂xi= 0. (3.7)

Note that if E is the standard Lie algebroid TM then the above equations arethe classical Euler-Lagrange equations for L : TM −→ R.

On the other hand, the Lagrangian function L is said to be regular if ωL is asymplectic section, that is, if ωL is regular at every point as a bilinear form. Insuch a case, there exists a unique solution ΓL verifying

iΓLΩL − dEL = 0 .

In addition, using (3.6), it follows that iSΓLωL = i∆ωL which implies that ΓL isa sode section, that is,

S(ΓL) = ∆,

or alternatively T τ(ΓL(a)) = a for all a ∈ E.Thus, the integral curves of ΓL (that is, the integral curves of the vector field

ρτ (ΓL)) are solutions of the Euler-Lagrange equations for L. ΓL is called theEuler-Lagrange section associated with L.

From (3.4), we deduce that L is regular if and only if the matrix Wαβ =∂2L

∂yα∂yβ

is regular. Moreover, the local expression of ΓL is

ΓL = yαXα + fαVα,

where the functions fα satisfy the linear equations

∂2L

∂yβ∂yαfβ +

∂2L

∂xi∂yαρi

βyβ +∂L

∂yγCγ

αβyβ − ρiα

∂L

∂xi= 0, for all α. (3.8)

Examples.1.- Real Lie algebras of finite dimension. Let g be a real Lie algebra of

finite dimension and L : g −→ R be a Lagrangian function. Then, the Euler-Lagrange equations for L are just the well-known Euler-Poincare equations(see, for instance, [36]).

2.- The tangent bundle. Let L : TM −→ R be a standard Lagrangian functionon the tangent bundle TM of M . Then, the resultant equations are the classicalEuler-Lagrange equations for L.

3.- Foliations. If the Lie algebroid is the tangent bundle of a foliation F on Pthen one recovers the classical formalism of holonomic mechanics.

4.- Atiyah algebroids. Let τQ|G : TQ/G −→ M be the Atiyah algebroidassociated with a principal G-bundle p : Q −→ M and L : TQ|G −→ R bea Lagrangian function. Then, the Euler-Lagrange equations for L are just theLagrange-Poincare equations (see [26]).

5.- Action Lie algebroids. Suppose that g is a real Lie algebra of finite di-mension and that Φ : g × V ∗ −→ V ∗ is a linear representation of g on V ∗. IfL : g×V ∗ −→ R is a Lagrangian function on the action Lie algebroid g×V ∗ −→ V ∗

then the Euler-Lagrange equations for L are just the so-called Euler-Poincare


equations with advected parameters or the Euler-Poisson-Poincare equa-tions (see [20]).

3.2. Hamiltonian Mechanics. In this section, we discuss how the Hamiltonianformalism can be developed for systems evolving on Lie algebroids (for more details,see [26, 39]).

Let τ∗ : E∗ −→ M be the vector bundle projection of the dual bundle E∗ to E.Consider the prolongation T EE∗ of E over τ∗,

T EE∗ = (b, v) ∈ E × TE∗ | ρ(b) = (Tτ∗)(v) = (a∗, b, v) ∈ E∗ × E × TE∗ | τ∗(a∗) = τ(b), ρ(b) = (Tτ∗)(v) .

The canonical geometrical structures defined on T EE∗ are the following:

• The Liouville section ΘE ∈ Sec((T EE∗)∗) defined by

ΘE(a∗)(b, v) = a∗(b). (3.9)

• The canonical symplectic section ΩE ∈ Sec(∧2(T EE∗)∗) is definedby

ΩE = −dΘE . (3.10)

where d is the differential on the Lie algebroid T EE∗.

Take coordinates (xi, pα) on E∗ and denote by Yα, Pβ the local basis of sectionsT EE∗, with

Yα(a∗) =(

a∗, eα(τ∗(a∗)), ρiα

∂

∂xi

)and Pβ(a∗) =

(a∗, 0,

∂

∂pα

).

In coordinates the Liouville and canonical symplectic sections are written as

ΘE = pαYα and ΩE = Yα ∧ Pα +12pγCγ

αβYα ∧ Yβ ,

where Yα,Pβ is the dual basis of Yα, Pβ.Every function H ∈ C∞(E∗) define a unique section ΓH of T EE∗ by the equation

iΓHΩE = dH,

and, therefore, a vector field ρτ∗(ΓH) = XH on E∗ which gives the dynamics. Incoordinates,

ΓH =∂H

∂pαYα −

(ρi

α

∂H

∂xi+ pγCγ

αβ

∂H

∂pβ

)Pα,

and therefore,

XH = ρiα

∂H

∂pα

∂

∂xi−

(ρi

α

∂H

∂xi+ pγCγ

αβ

∂H

∂pβ

)∂

∂pα.

Thus, the Hamilton equations are

dxi

dt= ρi

α

∂H

∂pα

dpα

dt= −ρi

α

∂H

∂xi− pγCγ

αβ

∂H

∂pβ. (3.11)

The Poisson bracket , ΩE defined by the canonical symplectic section ΩE onE∗ is the canonical Poisson bracket, which is known to exists on the dual of a Liealgebroid [3].


Examples.1.- Real Lie algebras of finite dimension. If the Lie algebroid E is a real

Lie algebra of finite dimension then the Hamilton equations are just the well-knownLie-Poisson equations (see, for instance, [36]).

2.- The tangent bundle. If E is the standard Lie algebroid TM and H :T ∗M −→ R is a Hamiltonian function then the resultant equations are the classicalHamilton equations for H.

3.- Foliations. If the Lie algebroid is the tangent bundle of a foliation F thenone recovers the classical formalism of holonomic Hamiltonian mechanics.

4.- Atiyah algebroids. Let τQ|G : TQ/G −→ M = Q|G be the Atiyah alge-broid associated with a principal G-bundle p : Q −→ M and H : T ∗Q|G −→ R bea Hamilton function. Then, the Hamilton equations for H are just the Hamilton-Poincare equations (see [26]).

5.- Action Lie algebroids. Suppose that g is a real Lie algebra of finite dimen-sion, that V is a real vector space of finite dimension and that Φ : g×V ∗ −→ V ∗ isa linear representation of g on V ∗. If H : g∗ × V ∗ −→ R is a Hamiltonian functionon the action Lie algebroid g× V ∗ −→ V ∗ then the Hamilton equations for H arejust the Lie-Poisson equations on the dual of the semidirect product ofLie algebras s = gsV (see [20]).

3.3. The Legendre transformation and the equivalence between the La-grangian and Hamiltonian formalisms. Let L : E −→ R be a Lagrangian func-tion and θL ∈ Sec((T EE)∗) be the Poincare-Cartan 1-section associated with L.

We introduce the Legendre transformation associated with L as the smoothmap LegL : E −→ E∗ defined by

LegL(a)(b) =d

dtL(a + tb)

∣∣t=0

, (3.12)

for a, b ∈ Ex, where Ex is the fiber of E over the point x ∈ M . In other wordsLegL(a)(b) = θL(a)(z), where z is a point in the fiber of T EE over the point a suchthat Tτ(z) = b.

The map LegL is well-defined and its local expression in fibred coordinates onE and E∗ is

LegL(xi, yα) = (xi,∂L

∂yα). (3.13)

From this local expression it is easy to prove that the Lagrangian L is regular ifand only if LegL is a local diffeomorphism.

The Legendre transformation induces a map T LegL : T EE −→ T EE∗ definedby

(T LegL)(b, Xa) = (b, (TaLegL)(Xa)), (3.14)

for a, b ∈ E and (a, b, Xa) ∈ T Ea E ⊆ Eτ(a) × Eτ(a) × TaE, where TLegL : TE −→

TE∗ is the tangent map of LegL. Note that τ∗ LegL = τ and thus T LegL iswell-defined.

If we consider local coordinates on T EE (resp. T EE∗) induced by the local basisXα, Vα (resp., Yα, Pα) the local expression of T LegL is

T LegL(xi, yα; zα, vα) = (xi,∂L

∂yα; zα, ρi

βzβ ∂2L

∂xi∂yα+ vβ ∂2L

∂yα∂yβ). (3.15)

The relationship between Lagrangian and Hamiltonian Mechanics is given bythe following result.


Theorem 3.1. [26] The pair (T LegL, LegL) is a morphism between the Lie al-gebroids (T EE, [[·, ·]]τ , ρτ ) and (T EE∗, [[·, ·]]τ∗ , ρτ∗). Moreover, if θL and ωL (re-spectively, ΘE and ΩE) are the Poincare-Cartan 1-section and 2-section associatedwith L (respectively, the Liouville 1-section and the canonical symplectic section onT EE∗) then

(T LegL, LegL)∗(ΘE) = θL, (T LegL, LegL)∗(ΩE) = ωL. (3.16)

In addition, in [26], it is proved that if the Lagrangian L is hyperregular, thatis, LegL is a global diffeomorphism, then (T LegL, LegL) is a symplectomorphismand the Euler-Lagrange section ΓL associated with L and the Hamiltonian sectionΓH are (T LegL, LegL)-related, that is,

ΓH LegL = T LegL ΓL. (3.17)

Therefore, an admissible curve a(t) on T EE is a solution of the Euler-Lagrangeequations if and only if the curve µ(t) = LegL(a(t)) is a solution of the Hamiltonequations.

4. Nonholonomic Lagrangian systems on Lie algebroids

4.1. Constrained Lagrangian systems. In this section, we will discuss La-grangian systems on a Lie algebroid τ : E −→ M subject to nonholonomic con-straints. The constraints are real functions on the positions and generalized veloci-ties which constrain the motion to some submanifold M of E. M is the constraintsubmanifold.

We will assume that the constraints are purely nonholonomic, that is, not all thegeneralized velocities are allowable, although all the positions are permitted. So,we will suppose that π = τ |M : M −→ M is a fibration.

The constraints are linear if they are linear functions on E or, in more geometricalterms, if M is a vector subbundle of E over M (Lagrangian systems subject to linearconstraints were discussed in [13, 45]).

In the general case, since π is a fibration, the prolongation T EM is defined.We will denote by r the dimension of the fibers of π : M −→ M , that is r =dim M− dim M .

Now, we define the bundle V −→ M of virtual displacements as the subbundleof τ∗E of rank r whose fiber at a point a ∈ M is

Va =

b ∈ Eτ(a)

∣∣ bV

a ∈ TaM

.

In other words, the elements of V are pairs of elements (a, b) ∈ E ⊕M E such thatd

dtφ(a + tb)

∣∣∣t=0

= 0,

for every local constraint function φ.We also define the bundle of constraint forces Ψ by Ψ = S∗((T EM)). Since

π is a fibration, the transformation S∗ : (T EM)0 −→ Ψ defines an isomorphismbetween the vector bundles (T EM)0 −→ M and Ψ −→ M. Therefore, the rank ofΨ is s = n− r, where n is the rank of E.

Next, suppose that L ∈ C∞(E) is a regular Lagrangian function. Then, the pair(L, M) is a constrained Lagrangian system. Moreover, assuming the validityof a Chetaev’s principle in the spirit of that of standard Nonholonomic Mechanics(see [25]), the solutions of the system (L, M) are curves t −→ c(t) on E such that:

– c is admissible (that is, ρ(c(t)) = m(t), where m = τ c),– c is contained in M and,


– i(c(t),c(t))ωL(c(t))− dEL(c(t)) ∈ Ψ(c(t)), for all t.

If (xi, yα) are local fibred coordinates on E, (ρiα, Cγ

αβ) are the correspondinglocal structure functions of E and

φA(xi, yα) = 0, A = 1, . . . , s,

are the local equations defining M as a submanifold of E, then∂φA

∂yαXα

A=1,...,s

is a local basis of Ψ. Moreover, a curve t −→ c(t) = (xi(t), yα(t)) on E is a solutionof the problem if and only if

xi = ρiαyα,

d

dt

(∂L

∂yα

)+

∂L

∂yγCγ

αβyβ − ρiα

∂L

∂xi= λA

∂φA

∂yα,

φA(xi, yα) = 0,

(4.1)

where λA are the Lagrange multipliers to be determined.These equations are called the Lagrange-d’Alembert equations for the con-

strained system (L,M). Note that if E is the Lie algebroid TM , then the aboveequations are just the standard Lagrange-d’Alembert equations for the constrainedsystem (L, M).

Now, we will assume that the solution curves of the problem are the integralcurves of a section Γ of T EE −→ E. Then, we may reformulate geometrically theproblem as follows: we look for a section Γ of T EE −→ E such that

(iΓωL − dEL)|M ∈ Sec(Ψ),

Γ|M ∈ Sec(T EM).(4.2)

If Γ is a solution of the above equations then, from (3.6), we have that

(iSΓωL − i∆ωL)|M = 0,

which implies that Γ is a sode section along M, that is, (SΓ−∆)|M = 0.

4.2. Regularity, projection of the free dynamics and nonholonomic bracket.We will discuss next the regularity of the constrained system (L,M) (the con-strained system (L,M) is regular if equations (4.2) admit a unique solution Γ).

For this purpose, we will introduce two new vector bundles F and T VM over M.The fibers of F and T VM at the point a ∈ M are

Fa = ω−1L (Ψa),

T Va M =

z ∈ T E

a M∣∣ T π(z) ∈ Va

=

z ∈ T E

a M∣∣ S(z) ∈ T E

a M

.

Then, one may prove the following result.

Theorem 4.1. [10] The following properties are equivalent:

(i) The constrained Lagrangian system (L, M) is regular.(ii) T EM ∩ F = 0.(iii) T VM ∩ (T VM)⊥ = 0.

Here, the orthogonal complement is taken with respect to the symplectic sec-tion ωL.

Condition (ii) (or, equivalently, (iii)) in Theorem 4.1 is locally equivalent to theregularity of the matrix

(CAB =

∂φA

∂yαWαβ ∂φB

∂yβ

)A,B=1,...,s


where (Wαβ) is the inverse matrix of(Wαβ =

∂2L

∂yα∂yβ

).

Thus, if L is a Lagrangian function of mechanical type (that is, L(a) = 12G(a, a)−

V (τ(a)), for all a ∈ E, with G : E×M E −→ R a bundle metric on E and V : M −→R a real function on M) then the constrained system (L,M) is always regular.

Now, assume that the constrained Lagrangian system (L, M) is regular. Then(ii) in Theorem 4.1 is equivalent to (T EE)|M = T EM ⊕ F and we will denote byP and Q the complementary projectors defined by this decomposition

Pa : T Ea E −→ T E

a M, Qa : T Ea E −→ Fa, for all a ∈ M.

Moreover, we have

Theorem 4.2. [10] Let (L, M) be a regular constrained Lagrangian system and letΓL be the solution of the free dynamics, i.e., iΓL

ωL = dEL. Then, the solution ofthe constrained dynamics is the sode Γ along M obtained as follows

Γ = P (ΓL|M).

On the other hand, (3) in Theorem 4.1 is equivalent to (T EE)|M = T VM ⊕(T VM)⊥ and we will denote by P and Q the corresponding projectors induced bythis decomposition, that is,

Pa : T Ea E −→ T V

a M, Qa = T Ea E −→ (T V

a M)⊥, for all a ∈ M.

Theorem 4.3. [10] Let (L, M) be a regular constrained Lagrangian system, ΓL

(respectively, Γ) be the solution of the free (respectively, constrained) dynamics and∆ be the Liouville section of T EE −→ E. Then, Γ = P (ΓL|M) if and only if therestriction to M of the vector field ρτ (∆) on E is tangent to M.

Note that if M is a vector subbundle of E then the vector field ρτ (∆) is tangentto M. Therefore, using Theorem 4.3, it follows that

Corollary 4.4. Under the same hypotheses as in Theorem 4.3 if M is a vectorsubbundle of E (that is, the constraints are linear) then Γ = P (ΓL|M).

Next, we will study the conservation of the Lagrangian energy for the constrainedLagrangian system (L, M).

Since S∗ : (T EM)0 −→ Ψ is a vector bundle isomorphism, it follows that thereexists a unique section α(L,M) of (T EM)0 −→ M such that

iQ(ΓL|M)ωL = S∗(α(L,M)).

Moreover, we have

Theorem 4.5 (Conservation of the energy). [10] If (L,M) is a regular constrainedLagrangian system and Γ is the solution of the dynamics then LΓ(EL|M) = 0 ifand only if α(L,M)(∆|M) = 0. In particular, if the vector field ρτ (∆) is tangent toM then LΓ(EL|M) = 0.

Now, suppose that f and g are two smooth real functions on M and take ar-bitrary extensions to E denoted by the same letters. Then, we may define thenonholonomic bracket of f and g as follows

f, gnh = ωL(P (Xf ), P (Xg))|M,

where Xf and Xg are the Hamiltonian sections on T EE associated with f and g,respectively.

The nonholonomic bracket is well-defined and, furthermore, it is not difficult toprove the following result.


Theorem 4.6 (The nonholonomic bracket). [10] The nonholonomic bracket is analmost-Poisson bracket, i.e., it is skew-symmetric and satisfies the Leibniz rule (itis a derivation in each argument with respect to the usual product of functions).Moreover, if f ∈ C∞(M) is an observable, then the evolution f of f is given by

f = ρτ (RL)(f) + f, EL|Mnh,

where RL is the section of T EM −→ M defined by RL = P (ΓL|M)− P (ΓL|M). Inparticular, if the vector field ρτ (∆) is tangent to M then

f = f,EL|Mnh.

4.3. Reduction. Next, we will discuss a reduction process and its relation with Liealgebroid epimorphisms. These results will be also valid for Lie algebroids withoutnonholonomic constraints, simply taking M = E and M′ = E′ in the sequel.

Let (L,M) be a regular constrained Lagrangian system on a Lie algebroid τ :E −→ M and (L′,M′) be another constrained Lagrangian system on a second Liealgebroid τ ′ : E′ −→ M ′. Suppose also that we have a fiberwise surjective morphismof Lie algebroids Φ : E −→ E′ over a surjective submersion φ : M −→ M ′ suchthat:

i) L = L′ Φ,ii) Φ|M : M −→ M′ is a surjective submersion and

iii) Φ(Va) = V′Φ(a), for all a ∈ M.

Note that if M and M′ are vector subbundles of E and E′, respectively, thenconditions i), ii) and iii) hold if and only if

L = L′ Φ and Φ(M) = M′.

In the general case, one may introduce the map T ΦΦ : T EM −→ T E′M′ given by

(T ΦΦ)(b, v) = (Φ(b), (TΦ)(v)), for (b, v) ∈ T EM,

and we have that T ΦΦ is a Lie algebroid epimorphism over Φ. In addition, thefollowing results hold

Theorem 4.7 (Reduction of the constrained dynamics). [10] Let (L,M) be a reg-ular constrained Lagrangian system on a Lie algebroid E and (L′, M′) be a con-strained Lagrangian system on a second Lie algebroid E′. Assume that we have afiberwise surjective morphism of Lie algebroids Φ : E −→ E′ over φ : M −→ M ′

such that conditions i), ii) and iii) hold. Then:

(i) The constrained Lagrangian system (L′,M′) is regular.(ii) If Γ (respectively, Γ′) is the constrained dynamics for L (respectively, for

L′) then T ΦΦ Γ = Γ′ Φ.(iii) If t −→ c(t) is a solution of Lagrange-d’Alembert differential equations for

L then Φ(c(t)) is a solution of Lagrange-d’Alembert differential equationsfor L′.

Theorem 4.8 (Reduction of the nonholonomic bracket). [10] Under the samehypotheses as in Theorem 4.7, we have that

f ′ Φ, g′ Φnh = f ′, g′′nh Φ

for f ′, g′ ∈ C∞(M′), where ·, ·nh (respectively, ·, ·′nh) is the nonholonomicbracket for the constrained system (L,M) (respectively, (L′,M′)). In other words,Φ : M −→ M′ is an almost-Poisson morphism when on M and M′ we consider thealmost-Poisson structures defined by the corresponding nonholonomic brackets.


Reduction by symmetries. Let φ : Q −→ M be a principal G-bundle and τ :E −→ Q be a Lie algebroid over Q. In addition, assume that we have an action ofG on E such that the quotient vector bundle E/G is defined and the set Sec(E)G

of equivariant sections of E is a Lie subalgebra of Sec(E). Then, E′ = E/Ghas a canonical Lie algebroid structure over M such that the canonical projectionΦ : E −→ E′ is a fiberwise bijective Lie algebroid morphism over φ (see [26]).

Next, suppose that (L, M) is a G-invariant regular constrained Lagrangian sys-tem, that is, the Lagrangian function L and the constraint submanifold M areG-invariant. Assume also that M is closed. Then, one may define a Lagrangianfunction L′ : E′ −→ R on E′ such that

L = L′ Φ.

Moreover, G acts on M and the set of orbits M′ = M/G of this action is a quotientmanifold, that is, M′ is a smooth manifold and the canonical projection Φ|M :M −→ M′ = M/G is a submersion. Thus, one may consider the constrainedLagrangian system (L′,M′) on E′.

Since the orbits of the action of G on E are the fibers of Φ and M is G-invariant,we deduce that

Va(Φ) ⊆ TaM, for all a ∈ M,

V (Φ) being the vertical bundle of Φ. This implies that Φ|Va: Va −→ V′Φ(a) is a

linear isomorphism, for all a ∈ M.

Therefore, from Theorem 4.7, we conclude that the constrained Lagrangian sys-tem (L′, M′) is regular and that

T ΦΦ Γ = Γ′ Φ,

where Γ (resp., Γ′) is the constrained dynamics for L (resp., L′). In addition, usingTheorem 4.8, we obtain that Φ : M −→ M′ is an almost-Poisson morphism when onM and M′ we consider the almost-Poisson structures induced by the correspondingnonholonomic brackets.

4.4. Example: a rolling ball on a rotating table. We apply the results in thissection to the case of a ball rolling without sliding on a rotating table with constantangular velocity [1, 4, 10, 30, 48]. A (homogeneous) sphere of radius r > 0, unitmass m = 1 and inertia k2 about any axis, rolls without sliding on a horizontaltable which rotates with constant angular velocity Ω about a vertical axis throughone of its points. Apart from the constant gravitational force, no other externalforces are assumed to act on the sphere.

Choose a Cartesian reference frame with origin at the center of rotation of thetable and z-axis along the rotation axis. Let (x, y) denote the position of the pointof contact of the sphere with the table. The configuration space for the sphere onthe table is Q = R2 × SO(3), where SO(3) may be parameterized by the Eulerianangles θ, ϕ and ψ. The kinetic energy of the sphere is then given by

T =12(x2 + y2 + k2(θ2 + ψ2 + 2ϕψ cos θ)).

With the potential energy being constant, we may put V = 0. Thus, the Lagrangianfunction L is T and the constraint equations are

x− rθ sinψ + rϕ sin θ cosψ = −Ωy,

y + rθ cos ψ + rϕ sin θ sin ψ = Ωx.

Since the Lagrangian function is of mechanical type, the constrained system isregular. Note that the constraints are not linear and that the restriction to the


constraint submanifold M of the Liouville vector field on TQ is not tangent to M.Indeed, the constraints are linear if and only if Ω = 0.

Next, following [4, 10], we will consider local coordinates (x, y, θ, ϕ, ψ; πi)i=1,...,5

on TQ = TR2 × T (SO(3)), where

x = x, y = y, θ = θ, ϕ = ϕ, ψ = ψ,

π1 = rx + k2q2, π2 = ry − k2q1, π3 = k2q3,

π4 =k2

(k2 + r2)(x− rq2 + Ωy), π5 =

k2

(k2 + r2)(y + rq1 − Ωx),

(q1, q2, q3) are the quasi-coordinates defined by

q1 = ωx, q2 = ωy, q3 = ωz,

and ωx, ωy and ωz are the components of the angular velocity of the sphere.Then, the constrained dynamics is the sode Γ along M defined by

Γ = (PΓL|M) = (x∂

∂x+ y

∂

∂y+ θ

∂

∂θ+ ϕ

∂

∂ϕ+ ψ

∂

∂ψ)|M

= (x∂

∂x+ y

∂

∂y+ q1

∂

∂q1+ q2

∂

∂q2+ q3

∂

∂q3)|M.

(4.3)

On the other hand, when constructing the nonholonomic bracket on M, we findthat the only non-zero fundamental brackets are

x, π1nh = r, y, π2nh = r,q1, π2nh = −1, q2, π1nh = 1, q3, π3nh = 1,

π1, π2nh = π3, π2, π3nh =k2

(k2 + r2)π1 +

rk2Ω(k2 + r2)

y,

π3, π1nh =k2

(k2 + r2)π2 − rk2Ω

(k2 + r2)x,

(4.4)

in which the “appropriate operational” meaning has to be attached to the quasi-coordinates qi.

Thus, we have that

f = RL(f) + f, Lnh, for f ∈ C∞(M),

where RL is the vector field on M given by

RL = (k2Ω

(k2 + r2)(x

∂

∂y− y

∂

∂x) +

rΩ(k2 + r2)

(x∂

∂q1+ y

∂

∂q2

+x(π3 − k2Ω)∂

∂π1+ y(π3 − k2Ω)

∂

∂π2− k2(π1x + π2y)

∂

∂π3))|M.

Note that RL = 0 if and only if Ω = 0.Now, it is clear that Q = R2×SO(3) is the total space of a trivial principal SO(3)-

bundle over R2 and the bundle projection φ : Q −→ M = R2 is just the canonicalprojection on the first factor. Therefore, we may consider the corresponding Atiyahalgebroid E′ = TQ/SO(3) over M = R2.

One may prove that E′ is isomorphic to the real vector bundle TR2×R3 −→ R2

in such a way that the anchor map ρ′ : E′ ∼= TR2×R3 −→ TR2 is just the canonicalprojection on the first factor. Moreover, one may choose a global basis e′ii=1,...,5

of Sec(E′) and the only non-zero fundamental Lie brackets are

[[e′4, e′3]]′ = e′5, [[e′5, e

′4]]′ = e′3, [[e′3, e

′5]]′ = e′4.

We have that the Lagrangian function L = T and the constraint submanifoldM are SO(3)-invariant. Consequently, L induces a Lagrangian function L′ onE′ = TQ/SO(3) and the set of orbits M′ = M/SO(3) is a submanifold of E′ =


TQ/SO(3) in such a way that the canonical projection Φ|M : M −→ M′ =M/SO(3) is a surjective submersion.

Under the identification between E′ = TQ/SO(3) and TR2 ×R3, L′ is given by

L′(x, y, x, y; ω1, ω2, ω3) =12(x2 + y2) +

k2

2(ω2

1 + ω22 + ω2

3),

where (x, y, x, y) and (ω1, ω2, ω3) are the standard coordinates on TR2 and R3,respectively. Moreover, the equations defining M′ as a submanifold of TR2 × R3

arex− rω2 + Ωy = 0, y + rω1 − Ωx = 0.

So, we have the constrained Lagrangian system (L′,M′) on the Atiyah algebroidE′ = TQ/SO(3) ∼= TR2 × R3. Note that the constraints are not linear and that if∆′ is the Liouville section of the prolongation T E′E′ then the restriction to M′ ofthe vector field (ρ′)τ ′(∆′) is not tangent to M′.

Now, if we put

x′ = x, y′ = y,π′1 = rx + k2ω2, π′2 = ry − k2ω1, π′3 = k2ω3,

π′4 =k2

(k2 + r2)(x− rω2 + Ωy), π′5 =

k2

(k2 + r2)(y + rω1 − Ωx),

then (x′, y′, π′1, π′2, π

′3, π

′4, π

′5) is a system of global coordinates on TQ/SO(3) ∼=

TR2 × R3. In these coordinates the equations defining the submanifold M′ areπ′4 = 0 and π′5 = 0 and the canonical projection Φ : TQ −→ TQ/SO(3) is given by

Φ(x, y, θ, ϕ, ψ; π1, π2, π3, π4, π5) = (x, y;π1, π2, π3, π4, π5). (4.5)

Thus, if Γ′ is the constrained dynamics for the system (L′, M′), it follows that(see (4.3))

(ρ′)τ ′(Γ′) = (x′∂

∂x′+ y′

∂

∂y′)|M′.

On the other hand, from (4.4), (4.5) and Theorem 4.8, we deduce that the onlynon-zero fundamental nonholonomic brackets for the system (L′,M′) are

x′, π′1′nh = r, y′, π′2′nh = r,

π′1, π′2′nh = π′3, π′2, π′3′nh =k2

(k2 + r2)π′1 +

rk2Ω(k2 + r2)

y′,

π′3, π′1′nh =k2

(k2 + r2)π′2 −

rk2Ω(k2 + r2)

x′.

Therefore, we have that

f ′ = (ρ′)τ ′(RL′)(f ′) + f ′, L′′nh, for f ′ ∈ C∞(M′),

where (ρ′)τ ′(RL′) is the vector field on M′ given by

(ρ′)τ ′(RL′) = k2Ωk2 + r2

(x′∂

∂y′− y′

∂

∂x′) +

rΩ(k2 + r2)

(x′(π′3 − k2Ω)∂

∂π′1+y′(π′3 − k2Ω)

∂

∂π′2− k2(π′1x

′ + π′2y′)

∂

∂π′3)|M′ .

4.5. Hamiltonian formalism. Let (L, M) be a constrained Lagrangian systemon a Lie algebroid E and assume that the Lagrangian function L is hyperregular.Then, since the LegL is a diffeomorphism then, it is clear that one may developa Hamiltonian formalism which is equivalent, via the Legendre transformation, tothe Lagrangian formalism.


5. Mechanical control systems on Lie algebroids

5.1. General control systems on Lie algebroids. Consider a Lie algebroidτ : E −→ M , with anchor map ρ : E −→ TM . Let σ, η1, . . . , ηk be sections of E.A control problem on the Lie algebroid τ : E −→ M with drift section σand input sections η1, . . . , ηk is defined by the following equation on M ,

m(t) = ρ(σ(m(t)) +

k∑

i=1

ui(t)ηi(m(t))), (5.1)

where u = (u1, . . . , uk) ∈ U , and U is an open set of Rk containing 0. The functiont 7→ u(t) = (u1(t), . . . , uk(t)) belongs to a certain class of functions of time, denotedby U, called the set of admissible controls. For our purposes, we may restrict theadmissible controls to be the piecewise constant functions with values in U . Noticethat the trajectories of the control system are admissible curves of the Lie algebroid,and therefore they must lie on a leaf of E. It follows that if E is not transitive,then there are points that cannot be connected by solutions of any control systemdefined on such a Lie algebroid. In particular, the system (5.1) cannot be locallyaccessible at points m ∈ M where ρ is not surjective. Since the emphasis here isput on the controllability analysis, without loss of generality we will restrict ourattention to locally transitive Lie algebroids.

Denoting by f = ρ(σ) and gi = ρ(ηi), i ∈ 1, . . . , k, we can rewrite the system (5.1)as

m(t) = f(m(t)) +k∑

i=1

ui(t)gi(m(t)), (5.2)

which is a standard nonlinear control system on M affine in the inputs [47]. Herewe make use of the additional geometric structure provided by the Lie algebroidin order to carry over the analysis of the controllability properties of the controlsystem (5.1). We refer to [47] for a comprehensive discussion of the notions ofreachable sets, accessibility algebra and computable accessibility tests.

Definition 5.1. The accessibility algebra D of the control system (5.1) inthe Lie algebroid is the smallest subalgebra of Sec(E) that contains the sectionsσ, η1, . . . , ηk.

Using the Jacobi identity, one can deduce that any element of accessibility alge-bra D is a linear combination of repeated Lie brackets of sections of the form

[[ζl, [[ζl−1, [[. . . , [[ζ2, ζ1]] . . .]]]]]],

where ζi ∈ σ, η1, . . . , ηk, 1 ≤ i ≤ l and l ∈ N.

Definition 5.2. The accessibility subbundle in the Lie algebroid, denotedby Lie(σ, η1, . . . , ηk), is the vector subbundle of E generated by the accessibilityalgebra D,

Lie(σ, η1, . . . , ηk) = span ζ(m) | ζ section of E in D , m ∈ M.

If the dimension of Lie(σ, η1, . . . , ηk) is constant, then Lie(σ, η1, . . . , ηk) isthe smallest Lie subalgebroid of E that has σ, η1, . . . , ηk as sections.

5.2. Mechanical control systems. Let τ : E −→ M be a Lie algebroid, let ∇ bea connection on E, and let η, η1, . . . , ηk be sections of E. A mechanical controlsystem on the Lie algebroid τ : E −→ M is defined by the following equation

∇a(t)a(t) + η(m(t)) =k∑

i=1

ui(t)ηi(m(t)). (5.3)


We will often refer to η as the potential energy term in equations (5.3). Associatedwith this equation, there is always a control system on the Lie algebroid T EE −→ Egiven by

a(t) = ρτ((Γ∇ − ηV )(a(t)) +

k∑

i=1

ui(t)ηV

i (a(t))), (5.4)

where ηV (resp. ηVi ) denotes the vertical lift of η (resp. ηi) and Γ∇ is the sode

section associated with ∇. Γ∇ is locally given by (see [13])

Γ∇ = yαXα − 12

(Γα

βγ + Γαγβ

)yβyγVα.

There are two distinguished families within the class of mechanical control sys-tems. We introduce them next.Mechanical control systems. Consider a Lagrangian system with L : E −→ Rof the form

L(a) =12G(a, a)− V τ(a), a ∈ E,

with G : E ×M E −→ R a bundle metric on E and V a function on M . ThisLagrangian function gives rise to the Euler-Lagrange equations as explained inSection 3.1.

Consider now the situation when the Lagrangian system is subject to some ex-ternal forces, represented by a collection θ1, . . . , θk of sections of E∗. Denote byη1, . . . , ηk the input sections of E determined by the control forces θ1, . . . , θk viathe metric, i.e., θi(X) = G(ηi, X) for all X ∈ Sec(E). If Γ∇G denotes the sode sec-tion associated with the Levi-Civita connection ∇G, the controlled Euler-Lagrangeequations can be written as

a(t) = ρτ(Γ∇G(a(t))− (gradG V )V (a(t)) +

k∑

i=1

ui(t)ηV

i (a(t))). (5.5)

Here gradG V is the section of E characterized by G(gradG V, X) = dV (X) forX ∈ Sec(E). Note that system (5.5) is a control problem on the Lie algebroidT EE −→ E as defined in Section 5.1. Locally, the equations can be written as

xi = ρiαyα,

yα = −12

(Γα

βγ(x) + Γαγβ(x)

)yβyγ − Gαβρi

β

∂V

∂xi+

k∑

i=1

ui(t)ηαi (x),

where (Gαβ) are the components of the metric G and (Gαβ) is the inverse matrix of(Gαβ).

Alternatively, one can describe the dynamical behavior of the mechanical controlsystem by means of an equation on E via the covariant derivative. An admissiblecurve a : t 7→ a(t) is a solution of the system (5.5) if and only if

∇Ga(t)a(t) + gradG V (m(t)) =

k∑

i=1

ui(t)ηi(m(t)). (5.6)

This equation corresponds to a mechanical control system (5.3) with connection∇ = ∇G and sections gradG V, η1, . . . , ηk.Mechanical control systems with constraints. Assume a mechanical controlsystem with data (G, V, θ1, . . . , θk) is subject to the constraints determined by asubbundle D of E. Consider the orthogonal decomposition E = D ⊕ D⊥ an the


associated orthogonal projectors P : E −→ D, Q : E −→ D⊥. Then, one can writethe controlled Lagrange-d’Alembert equations as

P (∇Ga(t)a(t)) + P (gradG V (m(t))) =

k∑

i=1

ui(t)P (ηi(m(t))), Q(a) = 0.

In terms of the constrained connection ∇ση = P (∇Gση) + ∇G

σ(Qη), with σ, η ∈Sec(E), the controlled equations can be rewritten as ∇a(t)a(t)+P (gradG V (m(t))) =∑k

i=1 ui(t)P (ηi(m(t))), Q(a) = 0. Since the forcing terms coming from the poten-tial and the inputs belong to D, the solutions of the total controlled dynamicsinitially belonging to D also remain in D. As a consequence, an admissible curvea : t 7→ a(t) is a solution of the system (5.8) if and only if

∇a(t)a(t) + P (gradG V (m(t))) =k∑

i=1

ui(t)P (ηi(m(t))), a0 ∈ D. (5.7)

This equation corresponds to a mechanical control system (5.3) with connection∇ = ∇ and sections P (gradG V ), P (η1), . . . , P (ηk).

Note that one can write the controlled dynamics as a control system on the Liealgebroid T EE −→ E,

a(t) = ρτ(Γ∇(a(t))− P (gradG V )V (a(t)) +

k∑

i=1

ui(t)P (ηi)V (a(t))). (5.8)

The coordinate expression of these equations is greatly simplified if we take a basiseα = ea, eA of E adapted to the orthogonal decomposition E = D ⊕ D⊥,i.e., D = spanea, D⊥ = spaneA. Denoting by (yα) = (ya, yA) the inducedcoordinates, the constraint equations Q(a) = 0 just read yA = 0. The controlledequations (5.7) are then

xi = ρiaya,

ya = −12Sa

bcybyc − Gaβρi

β

∂V

∂xi+

k∑

i=1

ui(t)P (ηi)a,

yA = 0.

where Sabc = Γb

ca + Γcba are the components of the symmetric product.

5.3. Accessibility and controllability notions. Here we introduce the notionsof accessibility and controllability that are specialized to mechanical control systemson Lie algebroids. Let m ∈ M and consider a neighborhood V of m in M . Definethe set of reachable points in the base manifold M starting from m as

RVM (m, T ) = m′ ∈ M | ∃u ∈ U defined on [0, T ] such that the evolution of (5.4)

for a(0) = 0m satisfies τ(a(t)) ∈ V, t ∈ [0, T ] and τ(a(T )) = m′ .

Alternatively, one may write RVM (m,T ) = τ(Rτ−1(V )

E (0m, T )). Denote

RVM (m,≤ T ) =

⋃

t≤T

RVM (m, t).

Definition 5.3. The system (5.4) is locally base accessible from m (respec-tively, locally base controllable from m) if RV

M (m,≤ T ) contains a non-emptyopen set of M (respectively, RV

M (m,≤ T ) contains a non-empty open set of M towhich m belongs) for all neighborhoods V of m and all T > 0. If this holds for anym ∈ M , then the system is called locally base accessible (respectively, locallybase controllable).


In addition to the notions of base accessibility and base controllability, we shallalso consider full-state accessibility and controllability starting from points of theform 0m ∈ E, m ∈ M (note that full-state is meant here with regards to E, not toTM).

Definition 5.4. The system (5.4) is locally accessible from m at zero (re-spectively, locally controllable from m at zero) if RW

E (0m,≤ T ) contains anon-empty open set of E (respectively, RW

E (0m,≤ T ) contains a non-empty openset of E to which 0m belongs) for all neighborhoods W of 0m in E and all T > 0.If this holds for any m ∈ M , then the system is called locally accessible at zero(respectively, locally controllable at zero).

The relevance of the above definitions stems from the fact that, frequently, oneneeds to control a system by starting at rest. Nevertheless it is important to noticethat not every equilibrium point at m corresponds to the point 0m. Finally, we alsointroduce the notion of accessibility and controllability with regards to a manifold.

Definition 5.5. Let ψ : M −→ N be an open mapping. The system (5.4) is lo-cally base accessible from m with regards to N (respectively, locally basecontrollable from m with regards to N) if ψ(RV

M (m,≤ T )) contains a non-emptyopen set of N (respectively, ψ(RV

M (m,≤ T )) contains a non-empty open set of Nto which ψ(m) belongs) for all neighborhoods V of m and all T > 0. If this holdsfor any m ∈ M , then the system is called locally base accessible with regardsto N (respectively, locally base controllable with regards to N).

Note that base accessibility and controllability with regards to M with idM :M −→ M corresponds to the notions of base accessibility and controllability (cf.Definition 5.3). Moreover, if the system is base accessible, then it is base accessiblewith regards to N . The analogous implication for base controllability also holdstrue.

5.4. The structure of the control Lie algebra. The aim of this section is toshow that the analysis of the structure of the control Lie algebra of affine connectioncontrol systems carried out in [29] can be further extended to control systemsdefined on a Lie algebroid. The enabling technical notion exploited here is that ofhomogeneity.

Let B be a Lie bracket formed with sections of the family X = Γ∇, ηV1 , . . . , ηV

k , ηV .The degree of B is the number of occurrences of all its factors, and is thereforegiven by δ(B) = δ0(B) + δ1(B) + · · · + δk(B), where δ0(B), δi(B), i ∈ 1, . . . , k,and δk+1(B) correspond, respectively, to the number of times that Γ∇, ηV

i , i ∈1, . . . , k, and ηV appear in B. For each l, consider the following sets

Brl(X) = B bracket in X | δ(B) = l, Brl(X) = B bracket in X | B ∈ Pl,where Pl denotes the set of homogeneous sections of T E of degree l. The notionof primitive bracket will also be useful. Given a bracket B in X, it is clear thatwe can write B = [B1, B2], with Bi brackets in X. In turn, we can also writeBα = [Bα1, Bα2] for α = 1, 2, and continue these decompositions until we end upwith elements belonging to X. The collection of brackets B1, B2, B11, B12, . . . arecalled the components of B. The components of B which do not admit furtherdecompositions are called irreducible. A bracket B is called primitive if all of itscomponents are brackets in Br−1(X) ∪ Br0(X) ∪ Γ∇.

Consider the set X′ = Γ∇ − ηV , ηV1 , . . . , ηV

k . Clearly, the elements in Lie(X′)are linear combinations of the elements in Lie(X). In fact, for each bracket B′

of elements in X′, let us define the subset S(B′) ⊂ Br(X) formed by all possible


brackets B ∈ Br(X) obtained by replacing each occurrence of Γ∇ − ηV in B′ byeither Γ∇ or ηV . Then, one can prove by induction (cf. [28]) that

B′ =∑

B∈S(B′)

(−1)δk+1(B)B. (5.9)

Reciprocally, given an element B ∈ Br(X), one can determine the bracket B′ ofelements in X′ such that B ∈ S(B′) simply by substituting each occurrence of Γ∇or ηV in B by Γ∇− ηV . We denote this operation by pseudoinv(B) = B′. For eachk ∈ N, define the following families of sections in E,

C(k)ver(η; η1, . . . , ηk) =

σ ∈ Sec(E) | σV = B′′, B′′ =∑

B∈S(pseudoinv(B))∩Br−1(X)∩Br0(X)

(−1)δk+1(B)B, B ∈ Br2k−1(X) primitive,

C(k)hor(η; η1, . . . , ηk) =

σ ∈ Sec(E) | σ = σB′′ , B′′ =

∑

B∈S(pseudoinv(B))∩Br−1(X)∩Br0(X)

(−1)δk+1(B)B, B ∈ Br2k(X) primitive.

Consider

Cver(η; η1, . . . , ηk) = ∪k∈NC(k)ver(η; η1, . . . , ηk),

Chor(η; η1, . . . , ηk) = ∪k∈NC(k)hor(η; η1, . . . , ηk),

and denote by Cver(η; η1, . . . , ηk) and Chor(η; η1, . . . , ηk), respectively, the subbun-dles of the Lie algebroid E generated by the latter families.

Taking into account the previous discussion, we are now ready to compute Lie(Γ∇−ηV , ηV

1 , . . . , ηV

k ) for a mechanical control system defined on a Lie algebroid.

Proposition 5.6. ([13]) Let m ∈ M . Then,

Lie(Γ∇ − ηV , ηV

1 , . . . , ηV

k ) ∩Ver0m(T E) = Cver(η; η1, . . . , ηk)(m)V ,

Lie(Γ∇ − ηV , ηV

1 , . . . , ηV

k ) ∩Horm(T E) = Chor(η; η1, . . . , ηk)(m).

Remark 5.7. In the absence of potential terms, i.e., η = 0, one has that

Cver(0; η1, . . . , ηk) = Sym(η1, . . . , ηk), Chor(0; η1, . . . , ηk) = Lie(Sym(η1, . . . , ηk)),where Sym(η1, . . . , ηk) denotes the distribution obtained by closing (the distri-bution defined by) η1, . . . , ηk under the symmetric product associated with ∇. Itis worth noticing that, in this case, Cver(0; η1, . . . , ηk) ⊆ Chor(0; η1, . . . , ηk). This isnot true in general. ¦

5.5. Accessibility and controllability tests. In this section we merge the no-tions introduced in Section 5.3 with the results obtained in Section 5.4 to give testsfor accessibility and controllability.

Proposition 5.8. [13] Let m ∈ M and assume the Lie algebroid E is locallytransitive at m. Then the mechanical control system (5.4) is

• locally base accessible from m if Chor(η; η1, . . . , ηk)(m) + ker ρ = Em,• locally accessible from m at zero if Chor(η; η1, . . . , ηk)(m) + ker ρ = Em

and Cver(η; η1, . . . , ηk)(m) = Em.

In order to state controllability tests, we need to introduce the notions of goodand bad symmetric products. We say that a symmetric product P in the sectionsη, η1, . . . , ηk is bad if the number of occurrences of each ηi in P is even. Otherwise,


P is good. Accordingly, 〈ηi : ηi〉 is bad and 〈〈η : ηj〉 : 〈ηi : ηi〉〉 is good. Thefollowing theorem gives sufficient conditions for local controllability.

Proposition 5.9. [13] Let m ∈ M . The mechanical control system (5.4) is

• locally base controllable from m if it is locally base accessible from m andevery bad symmetric product in η, η1, . . . , ηk evaluated at m can be putas an R-linear combination of good symmetric products of lower degree andelements of ker ρ,

• locally controllable from m at zero if it is locally accessible from m at zeroand every bad symmetric product in η, η1, . . . , ηk evaluated at m can beput as an R-linear combination of good symmetric products of lower degree.

The corresponding tests for base accessibility and controllability with regards toa manifold can be proved in a similar way.

Proposition 5.10. [13] Let ψ : M −→ N be an open map. Let m ∈ M and assumeψ∗(ρ(Em)) = Tψ(m)N . Then the mechanical control system (5.4) is

• locally base accessible from m with regards to N if Chor(η; η1, . . . , ηk)(m)+ρ−1(ker ψ∗) = Em,

• locally base controllable from m with regards to N if the system is locallybase accessible from m with regards to N and every bad symmetric productin η, η1, . . . , ηk evaluated at m can be put as an R-linear combination ofgood symmetric products of lower degree and elements of ρ−1(ker ψ∗).

6. Discrete Mechanics on Lie groupoids

In this section, we discuss discrete Lagrangian Mechanics on a Lie groupoidG ⇒ M . Instead of the usual Euler-Lagrange equations (3.7) for a Lie algebroidτ : E −→ M equipped with a Lagrange function L : E −→ R, we obtain aset of difference equations called Discrete Euler-Lagrange equations for a discreteLagrangian Ld : G −→ R [35]. When the Lie algebroid is precisely E = EG and Ld

is a suitable approximation of the continuous Lagrangian L : EG −→ R, then wewill obtain a geometric integrator for the Euler-Lagrange equations. In the nextsubsections we will carefully analyze this construction and its geometric properties.

6.1. Lie algebroid structure on the vector bundle πτ : EG

G×EG −→ G. LetG ⇒ M be a Lie groupoid with structural maps

α, β : G −→ M, ε : M −→ G, i : G −→ G, m : G2 −→ G.

We know that EG

G×EG ⇒ EG is a Lie groupoid. The following theorem shows that

the vector bundle πτ : EG

G× EG∼= V β ⊕G V α −→ G is equipped with a natural

structure of Lie algebroid.

Theorem 6.1 (See Theorem 3.3 in [35]). The vector bundle πτ : EG

G× EG∼=

V β ⊕G V α −→ G admits a Lie algebroid structure, where the anchor map is givenby

ρEG

G×EG(Xg, Yg) = Xg + Yg, for (Xg, Yg) ∈ Vgβ ⊕ Vgα, (6.1)

and the Lie bracket [[·, ·]]EG

G×EG on the space Sec(πτ ) is characterized by the followingrelation

[[(−→X,←−Y ), (

−→X ′,

←−Y ′)]]EG

G×EG =(−−−−−−→[[X, X ′]],

←−−−−[[Y, Y ′]]

), (6.2)

for X, Y, X ′, Y ′ ∈ Sec(τ).


We also remark that, if we denote by (G,α) the fibration α : G −→ M andby (G, β) the fibration β : G −→ M , then it is not difficult to prove that the Liealgebroid prolongations T EG(G,α) −→ G and T EG(G, β) −→ G are isomorphic, asLie algebroids, and both are isomorphic to the Lie algebroid V β⊕G V α −→ G and

hence to EG

G× EG −→ G (see [35] for the details).

The following diagram shows both structures of EG

G× EG,

EG

G× EG

ατ

βτ+3

ρEGG×EG

$$HHHH

HHHH

H

πτ

²²

EG

ρ

!!CCCC

CCCC

C

τ

²²

TGTα

Tβ+3

τG

yyssssssssssTM

τM||yyyy

yyyy

Gα

β+3 M

where the vertical maps are morphisms of Lie groupoids and the horizontal mapsare morphisms of Lie algebroids.

Given a section X of EG −→ M , we define the sections X(1,0), X(0,1) (the β and

α- lifts) and X(1,1) (the complete lift) of X to πτ : EG

G× EG −→ G as follows:

X(1,0)(g) = (−→X (g), 0g), X(0,1)(g) = (0g,

←−X (g)) and X(1,1)(g) = (−−→X (g),

←−X (g))

We can easily see that

[[X(1,0), Y (1,0)]]EG

G×EG = −[[X,Y ]](1,0)

[[X(0,1), Y (0,1)]]EG

G×EG = [[X,Y ]](0,1)and [[X(0,1), Y (1,0)]]EG

G×EG = 0 (6.3)

and, as a consequence,

[[X(1,1), Y (1,0)]]EG

G×EG = [[X, Y ]](1,0)

[[X(1,1), Y (0,1)]]EG

G×EG = [[X, Y ]](0,1)and [[X(1,1), Y (1,1)]]EG

G×EG = [[X, Y ]](1,1).

(6.4)

6.2. Discrete Variational Mechanics on Lie groupoids. Discrete Lagrangiansystems on Lie groupoids have a variational origin, as we explain next. A discreteLagrangian system consists of a Lie groupoid G ⇒ M (the discrete space) anda discrete Lagrangian Ld : G −→ R.Discrete Euler-Lagrange equations. For g ∈ G fixed, we consider the set ofadmissible sequences:

CNg =

(g1, . . . , gN ) ∈ GN

∣∣ (gk, gk+1) ∈ G2 for k = 1, . . . , N − 1 and g1 . . . gn = g

.

It is easy to show that we may identify the tangent space to CNg with

T(g1,...,gN )CNg ≡ (v1, . . . , vN−1) | vk ∈ (EG)xk

and xk = β(gk), 1 ≤ k ≤ N − 1 .

An element of T(g1,...,gN )CNg is called an infinitesimal variation. Now, we define

the discrete action sum associated to the discrete Lagrangian Ld : G −→ R by

SLd((g1, . . . , gN ) =N∑

k=1

Ld(gk).


Hamilton’s principle requires that this discrete action sum be stationary with re-spect to all the infinitesimal variations. This requirement gives the following alter-native expressions for the discrete Euler-Lagrange equations (see [35]):

←−X

(gk)(Ld)−−→X

(gk+1)(Ld) = 0, (6.5)

or

〈dLd, X(0,1)〉(gk)− 〈dLd, X

(1,0)〉(gk+1) = 0,

for all sections X of τ : EG −→ M . Here, d denotes the differential of the Lie

algebroid πτ : EG

G× EG ≡ V β ⊕G V α −→ G. Alternatively, we may rewrite theDiscrete Euler-Lagrange equations as

d[Ld lgk

+ Ld rgk+1 i](ε(xk))∣∣(EG)xk

= 0,

where β(gk) = α(gk+1) = xk, and where d denotes the standard differential on G,that is, the differential of the Lie algebroid τG : TG −→ G.

Thus, we may define the discrete Euler-Lagrange operator:

DDELLd : G2 −→ E∗G ,

where E∗G is the dual of EG. This operator is given by

DDELLd(g, h) = d [Ld lg + Ld rh i] (ε(x))∣∣(EG)xk

with β(g) = α(h) = x.

Discrete Poincare-Cartan sections. Consider the Lie algebroid πτ : EG

G×EG∼=

V β ⊕G V α −→ G, and define the Poincare-Cartan 1-sections Θ−Ld, Θ+

Ld∈

Sec((πτ )∗) as follows

Θ−Ld(g)(Xg, Yg) = −Xg(Ld), Θ+

Ld(g)(Xg, Yg) = Yg(Ld), (6.6)

for each g ∈ G and (Xg, Yg) ∈ Vgβ ⊕ Vgα.

Since dLd = Θ+Ld−Θ−Ld

and so, using d2 = 0, it follows that dΘ+Ld

= dΘ−Ld. This

means that there exists a unique 2-section ΩLd= −dΘ+

Ld= −dΘ−Ld

, that will becalled the Poincare-Cartan 2-section. This 2-section will be important to studythe symplectic character of the discrete Euler-Lagrange equations.

If Xi is a local basis of Sec(τ) then X(1,0)i , X

(0,1)i is a local basis of Sec(πτ ).

Moreover, if (Xi)(1,0), (Xi)(0,1) is the dual basis of X(1,0)i , X

(0,1)i , it follows that

Θ−Ld= −−→X i(Ld)(Xi)(1,0), Θ+

Ld=←−X i(Ld)(Xi)(0,1),

ΩLd= −−→X i(

←−X jLd)(Xi)(1,0) ∧ (Xj)(0,1).

Discrete Lagrangian evolution operator. Let ξ : G −→ G be a smooth mapsuch that:

- graph(ξ) ⊆ G2, that is, (g, ξ(g)) ∈ G2, for all g ∈ G (ξ is a second orderoperator).

- (g, ξ(g)) is a solution of the discrete Euler-Lagrange equations, for all g ∈G, that is, (DDELLd)(g, ξ(g)) = 0, for all g ∈ G.

In such case ←−X (g)(Ld)−−→X (ξ(g))(Ld) = 0 (6.7)

for every section X of EG and every g ∈ G. The map ξ : G −→ G is called adiscrete flow or a discrete Lagrangian evolution operator for Ld.


Now, let ξ : G −→ G be a second order operator. Then, the prolongationT ξ : V β⊕G V α −→ V β⊕G V α of ξ is the Lie algebroid morphism over ξ : G −→ Gdefined as follows (see [35]):

Tgξ(Xg, Yg) = ((Tg(rgξ(g)i))(Yg), (Tgξ)(Xg)+(Tgξ)(Yg)−Tg(rgξ(g)i)(Yg)), (6.8)

for all (Xg, Yg) ∈ Vgβ⊕Vgα. Moreover, from (2.7), (2.8) and (6.8), we obtain that

Tgξ(−→X (g),

←−Y (g)) = (−−→Y (ξ(g)), (Tgξ)(

−→X (g) +

←−Y (g)) +

−→Y (ξ(g))), (6.9)

for all X, Y sections of EG.Using (6.8), one may prove that (see [35]):

(i) The map ξ is a discrete Lagrangian evolution operator for Ld if and onlyif (T ξ, ξ)∗Θ−Ld

= Θ+Ld

.(ii) The map ξ is a discrete Lagrangian evolution operator for Ld if and only

if (T ξ, ξ)∗Θ−Ld−Θ−Ld

= dLd.(iii) If ξ is discrete Lagrangian evolution operator then (T ξ, ξ)∗ΩLd

= ΩLd.

Discrete Legendre transformations. Given a Lagrangian Ld : G −→ R wedefine the discrete Legendre transformations F−Ld : G −→ E∗

G and F+Ld :G −→ E∗

G by

(F−Ld)(h)(vε(α(h))) = −vε(α(h))(Ld rh i), for vε(α(h)) ∈ (EG)α(h),

(F+L)(g)(vε(β(g))) = vε(β(g))(Ld lg), for vε(β(g)) ∈ (EG)β(g).

Now, we introduce the prolongations T F−Ld : EG

G×EG ≡ V β⊕G V α −→ T EGE∗G

and T F+Ld : EG

G× EG ≡ V β ⊕G V α −→ T EGE∗G by

ThF−Ld(Xh, Yh) = (Th(i rh−1)(Xh), (ThF−L)(Xh) + (ThF−L)(Yh)),

ThF+Ld(Xh, Yh) = (Thlh−1(Yh), (ThF+L)(Xh) + (ThF+L)(Yh)),

for all h ∈ G and (Xh, Yh) ∈ Vhβ ⊕ Vhα. We observe that the discrete Poincare-Cartan 1-sections and 2-section are related to the canonical Liouville section of(T EGE∗

G)∗ −→ E∗G and the canonical symplectic section of ∧2(T EGE∗

G)∗ −→ E∗G

by pull-back under the discrete Legendre transformations, that is,

(T F−Ld,F−Ld)∗ΘEG= Θ−Ld

, (T F+Ld,F+Ld)∗ΘEG= Θ+

Ld,

(T F−Ld,F−Ld)∗ΩEG = ΩLd, (T F+Ld,F+Ld)∗ΩEG = ΩLd

.

Discrete regular Lagrangians. A discrete Lagrangian Ld : G −→ R is said to beregular if the set of solutions of the discrete Euler-Lagrange equations is locally thegraph of a diffeomorphism, that is, there exists locally a unique discrete Lagrangianevolution operator ξLd

: G −→ G for Ld. In such a case, ξLdis called the discrete

Euler-Lagrange evolution operator. In [35] (see Theorem 4.13 in [35]), we obtainedsome necessary and sufficient conditions for a discrete Lagrangian on a Lie groupoidG to be regular that we summarize as follows:

Ld is regular ⇐⇒ The Legendre transformation F+Ld is a local diffeomorphism⇐⇒ The Legendre transformation F−Ld is a local diffeomorphism⇐⇒ The Poincare-Cartan 2-section ΩLd

is symplectic

on the Lie algebroid EG

G× EG ≡ V β ⊕G V α −→ G.

Locally, we deduce that Ld is regular if and only if for every local basis Xi ofSec(τ) the local matrix (

−→X i(

←−X jLd)) is regular.


Discrete Hamiltonian evolution operator. If Ld : G −→ R is a regular La-grangian, then pushing forward to E∗

G with the discrete Legendre transformations,we obtain the discrete Hamiltonian evolution operator, ξLd

: E∗G −→ E∗

G

which is given byξLd

= F±Ld ξLd (F±Ld)−1 . (6.10)

Defining the prolongation T ξLd: T EGE∗

G −→ T EGE∗G of ξLd

by

T ξLd= T F±Ld T ξLd

(T F±Ld)−1,

we deduce that (see [35]):

(T ξLd, ξLd

)∗ΘEG= ΘEG

+ d(Ld (F−Ld)−1), (T ξLd, ξLd

)∗ΩEG= ΩEG

.

Noether’s theorem. In Discrete Mechanics is also possible to relate invarianceof the discrete Lagrangian under some transformation group with the existence ofconstants of the motion. In fact, we will say that a section X of EG is a Noether’ssymmetry of the Lagrangian Ld if there exists a function f ∈ C∞(M) such that

dLd(X(1,1)) = β∗f − α∗f.

In the particular case when dLd(X(1,1)) = −−→XLd +←−XLd = 0, we will say that X

is an infinitesimal symmetry of the discrete Lagrangian Ld.If Ld : G −→ R is a regular discrete Lagrangian, by a constant of the mo-

tion we mean a function F invariant under the discrete Euler-Lagrange evolutionoperator ξLd

, that is, F ξLd= F . Then, we have the following result.

Theorem 6.2 (Discrete Noether’s theorem). [35] If X is a Noether symmetry ofa discrete Lagrangian Ld, then the function F = Θ−Ld

(X(1,1))− α∗f is a constantof the motion for the discrete dynamics defined by Ld.

7. Classical Field Theory on Lie algebroids

In this section, we study Classical Field Theories on Lie algebroids. We considera fiber bundle ν : M −→ N , a Lie algebroid structure on a vector bundle τE

M : E −→M and a surjective morphism of Lie algebroids π : E −→ TN over ν. The physicalinterpretation of the above data is as follows: we will consider a field theory inwhich the fields are the sections of the bundle ν and the partial derivatives of thefields are parameterized by linear sections of π.

We will find the equations for the extremals of a variational problem whichroughly speaking is the following: given a Lagrangian function L defined on theset of sections of π, and a volume form ω on the manifold N , we look for thosemorphisms of Lie algebroids which are critical points of the action functional

S(Φ) =∫

N

L(Φ) ω.

This is a constrained variational problem, because we are restricting the fields Φ tobe morphisms of Lie algebroids, which is a condition on the derivatives of Φ.

7.1. Jets. We consider two vector bundles τEM : E −→ M and τF

N : F −→ N anda surjective vector bundle map π : E −→ F over the map ν : M −→ N . Moreover,we will assume that ν : M −→ N is a smooth fiber bundle. We will denote byK −→ M the kernel of the map π, which is a vector bundle over M . Given a pointm ∈ M , if we denote n = ν(m), we have the following exact sequence 0 −→ Km −→Em −→ Fn −→ 0, and we can consider the set Jmπ of splittings φ of such sequence.More concretely, we define the following sets Lmπ = w : Fn −→ Em | w is linear ,Jmπ = φ ∈ Lmπ | π φ = idFn and Vmπ = ψ ∈ Lmπ | π ψ = 0 . Therefore


Lmπ is a vector space, Vmπ is a vector subspace of Lmπ and Jmπ is an affinesubspace of Lmπ modeled on the vector space Vmπ. By taking the union, Lπ =∪m∈MLmπ, Jπ = ∪m∈MJmπ and Vπ = ∪m∈MVmπ, we get the vector bundleπ10 : Lπ −→ M and the affine subbundle π10 : Jπ −→ M modeled on the vectorbundle π10 : Vπ −→ M . We will also consider the projection π1 : Jπ −→ N definedby composition π1 = ν π10. An element of Jmπ will be simply called a jet at thepoint m ∈ M and accordingly the bundle Jπ is said to be the first jet bundle of π.

Notice that the standard case [52] is recovered when we have a bundle ν : M −→N and one considers the standard Lie algebroids E = TM −→ M and F = TN −→N together with the differential of the projection π = Tν : TM −→ TN . With thestandard notations, we have that J1ν ≡ J(Tν).

Local coordinates on Jπ are given as follows. We consider local coordinates(xi) on N and (xi, uA) on M adapted to the projection ν. We also consider localbasis of sections ea of F and ea, eα of E adapted to the projection π, that isπ ea = ea ν and π eα = 0. In this way eα is a base of sections of K. Anelement w in Lmπ is of the form w = (wb

aeb + wαa eα)⊗ ea, and it is in Jmπ if and

only if wba = δb

a, i.e., an element φ in Jπ is of the form φ = (ea + φαaeα)⊗ ea. If we

set yαa (φ) = φα

a , we have adapted local coordinates (xi, uA, yαa ) on Jπ. Similarly,

an element ψ ∈ Vmπ is of the form ψ = ψαa eα ⊗ ea. If we set yα

a (ψ) = ψαa , we have

adapted local coordinates (xi, uA, yαa ) on Vπ. As usual, we use the same name for

the coordinates in an affine bundle and in the associated vector bundle.An element z ∈ L∗mπ defines an affine function z on Jmπ by contraction z(φ) =

〈z, φ〉 where 〈·, ·〉 is the pairing 〈z, φ〉 = Tr(zφ) = Tr(φz). Therefore, a section θ ofL∗π defines a fiberwise affine function θ on Jπ, θ(φ) = 〈θπ10(φ), φ〉 = Tr(θπ10(φ) φ).In local coordinates, a section of L∗π is of the form θ = (θa

b (x)eb + θaα(x)eα) ⊗ ea,

and the affine function defined by θ is θ = θaa(x) + θa

α(x)yαa .

Anchor. Consider now anchored structures on the bundles E and F , that is, wehave two vector bundle maps ρF : F −→ TN and ρE : E −→ TM over the identityin N and M respectively. We will assume that the map π is admissible, that isρF π = Tν ρE . Therefore we have

ρF (ea) = ρia

∂

∂xiand

ρE(ea) = ρia

∂

∂xi+ ρA

a

∂

∂uA,

ρE(eα) = ρAα

∂

∂uA,

with ρia = ρi

a(x), ρAa = ρA

a (x, u) and ρAα = ρA

α (x, u).The anchor allows us to define the concept of total derivative of a function with

respect to a section. Given a section σ ∈ Sec(F ), the total derivative of a functionf ∈ C∞(M) with respect to σ is the function df ⊗ σ, i.e., the affine functionassociated to df ⊗ σ ∈ Sec(L∗π). In particular, the total derivative with respect toan element ea of the local basis of sections of F , will be denoted by f|a. In thisway, if σ = σaea then df ⊗ σ = f|aσa, where the coordinate expression of f|a is

f|a = ρia

∂f

∂xi+ (ρA

a + ρAαyα

a )∂f

∂uA.

Notice that, for a function f in the base N , we have that f|a = ρia

∂f∂xi are just the

components of df in the basis ea.Bracket. Finally, let us assume that we have Lie algebroid structures on τF

N : F −→N and on τE

M : E −→ M , and that the projection π is a morphism of Lie algebroids.This condition implies the vanishing of some structure functions.


We have the following expressions for the brackets of elements in the basis ofsections

[[ea, eb]] = Ccabec and

[[ea, eb]] = Cγabeγ + Cc

abec

[[ea, eβ [[= Cγaβeγ

[[eα, eβ ]] = Cγαβeγ ,

where Cabc = Ca

bc(x) is a basic function.The structure functions can be conveniently combined as terms of some affine

function as follows

Zαaγ = Cα

aγ + Cαβγyβ

a and Zαac = Cα

ac + Cαβcy

βa .

In particular, such functions will appear in the Euler-Lagrange equations of thevariational problem.

7.2. Morphisms and admissible maps. By a section of π we mean a vectorbundle map Φ such that π Φ = idF , (i.e., we consider only linear sections of π(see also [16])). It follows that the base map φ : N −→ M is a section of ν, i.e.,ν φ = idN . The set of sections of π will be denoted by Sec(π). The set of thosesections of π which are a morphism of Lie algebroids will be denoted by M(π). Wewill find local conditions for Φ ∈ Sec(π) to be an admissible map between anchoredvector bundles and also local conditions for Φ to be a morphism of Lie algebroids.

Taking adapted local coordinates (xi, uA) on M , the map φ has the expressionφ(xi) = (xi, uA(x)). If we moreover take an adapted basis ea, eα of local sectionsof E, then the expression of Φ is given by Φ(ea) = ea + yα

a (x)eα, so that the map Φis determined by the functions

(uA(x), yα

a (x))

locally defined on N . The action onthe dual basis is Φ?ea = ea, and Φ?eα = yα

a (x)ea, and for the coordinate functionsΦ?xi = xi and Φ?uA = uA(x).

The admissibility condition reads Φ?(df) = d(Φ?f) for every function f ∈C∞(M). Taking f = xi we get an identity, while taking f = uA we get thecondition

ρia

∂uA

∂xi= ρA

a + ρAαyα

a .

In addition to the admissibility condition, the morphism condition reads Φ?dθ =d(Φ?θ) for every section θ of E∗. For θ = ea we get an identity, while for θ = eα

we find the

ρib

∂yαc

∂xi− ρi

c

∂yαb

∂xi− yα

a Cabc + Cα

βγyβb yγ

c + Cαbγyγ

c − Cαcγyγ

b + Cαbc = 0.

7.3. Variational Calculus. In what follows in this paper we consider the casewhere the Lie algebroid F is the tangent bundle F = TN with ρF = idTN and [·, ·]the usual Lie bracket of vector fields on N . The Lie algebroid E remains a generalLie algebroid. Moreover, for local expressions on F , the local basis of sections ofF which we will consider is a basis of coordinate vector fields ei = ∂

∂xi , so thatρi

a = δia and Ca

bc = 0.Variational problem. Given a Lagrangian function L ∈ C∞(Jπ) and a volumeform ω ∈ ∧r (TN), where r = dim(N), we consider the following variational prob-lem: find the critical points of the action functional S(Φ) =

∫N

L(Φ) ω defined onthe set of sections of π which are moreover morphisms of Lie algebroids, that is,defined on the set M(π). Here by L(Φ) we mean the function n 7→ L(Φn), whereΦn ∈ Jπ is the restriction of Φ the fiber Fn = TnN .

It is important to notice that the above variational problem is a constrainedproblem, not only because the condition π Φ = idF , which can be easily solved,


but because of the condition of Φ being a morphism of Lie algebroids, which is acondition on the derivatives of Φ. Taking coordinates on N such that the volumeform is ω = dx1 ∧ · · · ∧ dxr, the problem is to find the critical points of∫

N

L(xi, uA(x), yαa (x)) dx1 ∧ · · · ∧ dxr,

subject to the constraints

∂uA

∂xa= ρA

a + ρAαyα

a and∂yα

c

∂xb− ∂yα

b

∂xc+ Cα

bγyγc − Cα

cγyγb + Cα

βγyβb yγ

c + Cαbc = 0.

The first method one can try to solve the problem is to use Lagrange multipliers.Nevertheless, one has no warranties that all solutions to this problem are normal(i.e., not strictly abnormal). In fact, in simple cases such as the problem of a heavytop [38], one can easily see that there will be strictly abnormal solutions. Thereforewe take another approach, which consists of finding explicitly finite variations of asolution, that is, defining a curve in M(π) starting at the given solution.Variations and infinitesimal variations. In order to find admissible variations,we consider sections of E and the associated flow. With the help of this flow wecan transform morphisms of Lie algebroids into morphisms of Lie algebroids.Flow defined by a section. We recall that every section of a Lie algebroid hasan associated local flow composed of morphisms of Lie algebroids [42, 34]. Moreexplicitly, given a section σ of a Lie algebroid E, there exists a local flow Φs : E −→E such that

Lσθ =d

dsΦ?

sθ∣∣∣s=0

,

for any section θ of∧

E. Moreover, for every fixed s, the map Φs is a morphism ofLie algebroids, and the base map φs : M −→ M , the (ordinary) flow of the vectorfield ρ(σ) ∈ X(M).Complete lift of a section. In this section we will define the lift of a projectablesection of E to a vector field on Jπ, in a similar way to the definition of the first jetprolongations of a projectable vector field in the standard theory of jet bundles [52].

We consider a section σ of a Lie algebroid E projectable over a section η of F .We denote by Ψs the flow on E associated to σ and by Φs the flow on F associatedto η. We recall that, for every fixed s, the maps Ψs and Φs are morphisms of Liealgebroids. Moreover, the base maps ψs and φs, are but the flows of the vectorfields ρE(σ) and ρF (η), respectively.

The projectability of the section implies the projectability of the flow. It followsthat (locally, in the domain of the flows) we have defined a map LΨs : Lπ −→ Lπby means of LΨs(w) = Ψs w Φ−s. By restriction of LΨs to Jπ we get a mapJΨs, which is a local flow in Jπ. We will denote by X(1)

σ the vector field on Jπgenerating the flow JΨs. The vector field X(1)

σ will be called the complete lift toJπ of the section σ. Since JΨs projects to the flow ψs it follows that the vectorfield X(1)

σ projects to the vector field ρE(σ) in M .Locally, a section σ = σaea + σαeα is projectable if σa = σa(xi) depends only

on xi. Its complete lift X(1)σ has the local expression

X(1)σ = σa ∂

∂xa+ (ρA

a σa + ρAασα)

∂

∂uA+ σα

a

∂

∂yαa

,

where σαa = σα

|a + Zαabσ

b + Zαaβσβ − yα

b

(σb|a + σcCb

ac

). In particular, if σ projects to

the zero section, i.e., σa = 0, we have

X(1)σ = ρA

ασα ∂

∂uA+

(σα|a + Zα

aβσβ) ∂

∂yαa

.


Euler-Lagrange equations. Let Φ ∈ M(π) be a critical point of S. In orderto find admissible variations we consider a π-vertical section σ of E. Its flowΨs : E −→ E projects to the identity in F = TN . Moreover we will require σ tohave compact support. Since for every fixed s, the map Ψs is a morphism of Liealgebroids, it follows that the map Φs = Ψs Φ is a section of π and a morphismof Lie algebroids, that is, s 7→ Φs is a curve in M(π). Using this kind of variationswe have the following result.

Theorem 7.1. [42] Select local coordinates such that the volume form is expressedas ω = dx1 ∧ · · · ∧ dxr. A map Φ is a critical section of S if and only if thecomponents yα

a of Φ satisfy the system of partial differential equations

∂uA

∂xa= ρA

a + ρAαyα

a ,

∂yαa

∂xb− ∂yα

b

∂xa+ Cα

bγyγa − Cα

aγyγb + Cα

βγyβb yγ

a + Cαba = 0,

d

dxa

(∂L

∂yαa

)=

∂L

∂yγa

Zγaα +

∂L

∂uAρA

α .

Proof. Recall that by L(Φ) we mean the function in N given by L(Φ)(n) = L(Φn),where Φn is the restriction of Φ: F −→ E to the fiber Fn. The function L(Φs) is

L(Φs)(n) = L(Ψs Φn) = L(JΨs(Φn)) = (JΨ∗sL)(Φ)(n),

and therefore the variation of the action along the curve s 7→ Φs is

0 =d

dsS(Φs)

∣∣∣s=0

=∫

N

d

dsL(Φs)

∣∣∣s=0

ω =∫

N

(LX

(1)σ

L)(Φ)ω.

Taking into account the local expression of X(1)σ for a π-vertical σ, we have that

LX

(1)σ

L = ρAασα ∂L

∂uA+

(dσα

dxa+ Zα

aβσβ

)∂L

∂yαa

= σα

[ρA

α

∂L

∂uA+ Zγ

aα

∂L

∂yγa− d

dxa

(∂L

∂yαa

)]+

d

dxa

(σα ∂L

∂yαa

).

Let us denote by δL the expression with components

δLα =d

dxa

(∂L

∂yαa

)− Zγ

aα

∂L

∂yγa− ρA

α

∂L

∂uA,

and by Jσ the (r − 1)-form (along π1) Jσ = σα ∂L∂yα

aωa with ωa = i ∂

∂xaω. Then we

have that0 =

d

dsS(Φs)

∣∣∣s=0

= −∫

N

(δLα σα)ω +∫

N

d(Jσ Φ).

Since σ has compact support the second term vanishes by Stokes theorem. More-over, since the section σ is arbitrary, by the fundamental theorem of the Calculusof Variations, we get δL = 0, which are the Euler-Lagrange equations. Noticethat the first two equations in the above statement are nothing but the morphismconditions. ¤

7.4. Examples.

Standard case. In the standard case, we consider a bundle ν : M −→ N , the stan-dard Lie algebroids F = TN and E = TM and the tangent map π = Tν : TM −→TN . Then we have that Jπ = J1ν. When we choose coordinate basis of vectorfields (i.e., of sections of TN and TM) we recover the equations for the standardfirst-order field theory. Moreover, if we consider a different basis, what we get arethe equations for a first-order field theory written in pseudo-coordinates [5, 14, 42].


Time-dependent Mechanics. In [50, 51] we developed a theory of Lagrangian Me-chanics for time dependent systems defined on Lie algebroids, where the base man-ifold is fibered over the real line R. Since time-dependent Mechanics is nothing buta 1-dimensional field theory, our results must reduce to that.

The morphism condition is just the admissibility condition so that, if we writex0 ≡ t and yα

0 ≡ yα, the Euler-Lagrange equations are

duA

dt= ρA

0 + ρAαyα,

d

dt

(∂L

∂yα

)=

∂L

∂yγ(Cγ

0α + Cγβαyβ) +

∂L

∂uAρA

α ,

in full agreement with [44]. In particular, for an autonomous system on a Liealgebroid V −→ Q, one considers E = TR×V −→ R×M with π the projection ontothe first factor TR. Our results provide yet another indication of the variationalcharacter of autonomous mechanical systems on Lie algebroids.

Topological field theories. Given a closed r-form Ω on a Lie algebroid V −→ Q,we can define a topological field theory as follows. For an r-dimensional manifoldN we consider F = TN −→ N , E = TN × V −→ N × Q and π the projectiononto the first factor TN . The Lagrangian of the theory is L(Φ) = Φ?Ω. Then it iseasy to see that the Euler-Lagrange equations reduce to the morphism condition.In this way, one can cope with systems such as Poisson σ-models or Chern-Simonstheories [2, 42, 43].

Systems with symmetry. The case of a system with symmetry is very importantin Physics. We consider a principal bundle ν : P −→ M with structure group Gand we set N = M , F = TN and E = TP/G (the Atiyah algebroid of P ), withπ([v]) = Tν(v). Sections of π are just principal connections on P and a section isa morphism if and only if it is a flat connection. The kernel K is just the adjointbundle (P×g)/G −→ M . By an adequate choice of a local basis of sections of F , Kand E one easily find the covariant Euler-Poincare equations [7, 8]. The covariantLagrange-Poincare equations [6] can also be recovered within this formalism.

8. Future work

We have illustrated the generality of the theory of Lie algebroids and groupoidsin a wide range of situations, from nonholonomic Lagrangian and Hamiltoniansystems and mechanical control systems to Discrete Mechanics and extensions toField Theory. Current and future directions of research include the following:

Hamilton-Jacobi equation for a Hamiltonian system on a Lie algebroid: It wo-uld be interesting to continue with the study started in [26] of Hamilton-Jacobi theory for Hamiltonian systems on Lie algebroids. In particular, itwould be interesting to introduce a suitable definition of a local (global)complete integral of the Hamilton-Jacobi equation. The idea would bethat the knowledge of an integral of the equation would allow the “directdetermination” of some integral curves of the corresponding Hamiltonianvector field.

Geometric formalism for Vakonomic Mechanics on Lie algebroids: An inter-esting topic to study is the case of constrained variational problems on Liealgebroids. In this case, to derive the equations of motion for a Lagrangiansystem subject to nonholonomic constraints, one invokes a variational prin-ciple, rather than the Lagrange-D’Alembert’s principle (cf. Section 4). Thedifferential equations obtained, called vakonomic equations, are in generaldifferent. From an optimal control perspective, it seems interesting togeneralize the formalism developed in [11].


Mechanical control systems on Lie algebroids: Topics of interest related tomechanical control systems on Lie algebroids include the investigation ofcontrollability tests along relative equilibria and the study of systems thatinclude gyroscopic and dissipative forces.

Discrete Mechanics on Lie groupoids: We are currently studying the construc-tion of geometric integrators for mechanical systems on Lie algebroids.We have introduced the exact discrete Lagrangian in the Lie groupoidformalism, and are discussing different types of discretizations of continu-ous Lagrangians and their numerical implementation. We plan to explorenatural extensions to forced systems and to systems with holonomic andnonholonomic constraints as in [12, 27].

Classical Field Theory and Lie algebroids: In [16, 44] the authors have intro-duced the notion of a Lie affgebroid structure [17, 41, 44] (see also [21]).They have developed a Lagrangian (and Hamiltonian) formalism on Lie af-fgebroids, which generalizes some classical formalisms for time-dependentMechanics and, in addition, may be applied to other situations. Sincetime-dependent Mechanics is a 1-dimensional field theory, it would be in-teresting to define the notion of a “Lie multialgebroid”, as a generalizationof the notion of a Lie affgebroid. This mathematical object should en-code the geometric structure necessary to develop field theories. The firstexample of a Lie multialgebroid¡ should be Jπ. The notion of a Lie multi-algebroid will potentially allow to study other aspects of the theory, suchas Tulczyjew’s triples associated with a Lie multialgebroid and Hamilton-Jacobi equation for classical field theories on Lie multialgebroids.

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Jorge Cortes: Applied Mathematics and Statistics, Baskin School of Engineering,University of California, Santa Cruz, California 95064, USA

E-mail address: [email protected]

M. de Leon: Instituto de Matematicas y Fısica Fundamental, Consejo Superior deInvestigaciones Cientıficas, Serrano 123, 28006 Madrid, Spain


Juan C. Marrero: Departamento de Matematica Fundamental, Facultad de Matema-ticas, Universidad de la Laguna, La Laguna, Tenerife, Canary Islands, Spain


D. Martın de Diego: Instituto de Matematicas y Fısica Fundamental, Consejo Supe-rior de Investigaciones Cientıficas, Serrano 123, 28006 Madrid, Spain


Eduardo Martınez: Departamento de Matematica Aplicada, Facultad de Ciencias,Universidad de Zaragoza, 50009 Zaragoza, Spain


a survey of lagrangian mechanics and control on lie

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